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G.H.HARDY /Foreword by C.P. Snow 




"It was a perfectly ordinary night at Christ's 
high table, except that Hardy was' dining as a 
guest. He had just returned to Cambridge as 
Sadleirian professor, and I had heard some- 
thing of him from young Cambridge mathe- 
maticians. They were delighted to have him 
back: he was a real mathematician, they said, 
not like those Diracs and Bohrs the physicists 
were always talking about: he was the purest 
of the pure. He was also unorthodox, eccentric, 
radical, ready to talk about anything." 

So writes C. P. Snow in his introduction to 
this reissue of a book that has long been treas- 
ured by mathematicians, and others to whom 
mathematics is an attractive mystery. 

The book is a personal account by a distin- 
guished mathematician of what mathematics 
meant to him as a man. Hardy discusses and 
illustrates the attractive force of mathematics. 
He dismisses its utility but describes its depth 
and beauty as a creative art. 

Hardy asks : Is it worth man's while to give 
his life to the study of mathematics ? His an- 
swer is of necessity personal, and indeed that 
is the book's value. It holds the confessions of 
an unrepentant but humane mathematician; 
his candid integrity and good humour make a 
finite addition to the sum of human thought. 


15s net 
in U.K. 








Published by the Syndics of the Cambridge University Press 

Bentley House, 200 Euston Road, London, N.W. 1. 

American Branch: 32 East 57th Street, New York, N.Y. 10022 

Foreword © C. P. Snow 1967 

First Edition 1940 

Reprinted 194.1 


Reprinted with Foreword 
by G. P. Snow 1967 


who asked me to write it 

Printed by offset lithography in the United States 
of America 


It was a perfectly ordinary night at Christ's high 
table, except that Hardy was dining as a guest. He 
had just returned to Cambridge as Sadleirian 
professor, and I had heard something of him from 
young Cambridge mathematicians. They were 
delighted to have him back : he was a real mathe- 
matician, they said, not like those Diracs and 
Bohrs the physicists were always talking about : he 
was the purest of the pure. He was also unorthodox, 
eccentric, radical, ready to talk about anything. 
This was 1931, and the phrase was not yet in 
English use, but in later days they would have 
said that in some indefinable way he had star 

So, from lower down the table, I kept studying 
him. He was then in his early fifties : his hair was 
already grey, above skin so deeply sunburnt that it 
stayed a kind of Red Indian bronze. His face 
was beautiful — high cheek bones, thin nose, 
spiritual and austere but capable of dissolving into 
convulsions of internal gamin-like amusement. He 
had opaque brown eyes, bright as a bird's — a 
kind of eye not uncommon among those with a 
gift for conceptual thought. Cambridge at that 
time was full of unusual and distinguished faces — 

but even then, I thought that night, Hardy's 
stood out. 

I do not remember what he was wearing. It 
may easily have been a sports coat and grey 
flannels under his gown. Like Einstein, he dressed 
to please himself: though, unlike Einstein, he 
diversified his casual clothing by a taste for 
expensive silk shirts. 

As we sat round the combination-room table, 
drinking wine after dinner, someone said that 
Hardy wanted to talk to me about cricket. I had 
been elected only a year before, but Christ's was 
then a small college, and the pastimes of even the 
junior fellows were soon identified. I was taken to 
sit by him. I was not introduced. He was, as I 
later discovered, shy and self-conscious in all 
formal actions, and had a dread of introductions. 
He just put his head down as it were in a butt of 
acknowledgment, and without any preamble what- 
ever began: 

'You're supposed to know something about 
cricket, aren't you?' Yes, I said, I knew a bit. 

Immediately he began to put me through a 
moderately stiff viva. Did I play? What sort of 
performer was I? I half-guessed that he had a 
horror of persons, then prevalent in academic 
society, who devotedly studied the literature but 
had never played the game. I trotted out my 
credentials, such as they were. He appeared to 


find the reply partially reassuring, and went on to 
more tactical questions. Whom should I have 
chosen as captain for the last test match a year 
before (in 1930)? If the selectors had decided that 
Snow was the man to save England, what would 
have been my strategy and tactics? ('You are 
allowed to act, if you are sufficiently modest, as 
non-playing captain. ') And so on, oblivious to the 
rest of the table. He was quite absorbed. 

As I had plenty of opportunities to realize in 
the future, Hardy had no faith in intuitions or 
impressions, his own or anyone else's. The only- 
way to assess someone's knowledge, in Hardy's 
view, was to examine him. That went for mathe- 
matics, literature, philosophy, politics, anything 
you like. If the man had bluffed and then wilted 
under the questions, that was his lookout. First 
things came first, in that brilliant and concentrated 

That night in the combination-room, it was 
necessary to discover whether I should be tolerable 
as a cricket companion. Nothing else mattered. In 
the end he smiled with immense charm, with 
child-like openness, and said that Fenner's (the 
university cricket ground) next season might be 
bearable after all, with the prospect of some 
reasonable conversation. 

Thus, just as I owed my acquaintanceship with 
Lloyd George to his passion for phrenology, I 


owed my friendship with Hardy to having wasted 
a disproportionate amount of my youth on 
cricket. I don't know what the moral is. But it was 
a major piece of luck for me. This was intellec- 
tually the most valuable friendship of my life. His 
mind, as I have just mentioned, was brilliant and 
concentrated : so much so that by his side anyone 
else's seemed a little muddy, a little pedestrian and 
confused. He wasn't a great genius, as Einstein 
and Rutherford were. He said, with his usual 
clarity, that if the word meant anything he was 
not a genius at all. At his best, he said, he was for a 
short time the fifth best pure mathematician in 
the world. Since his character was as beautiful and 
candid as his mind, he always made the point that 
his friend and collaborator Littlewood was an 
appreciably more powerful mathematician than he 
was, and that his protege Ramanujan really had 
natural genius in the sense (though not to the 
extent, and nothing like so effectively) that the 
greatest mathematicians had it. 

People sometimes thought he was under-rating 
himself, when he spoke of these friends. It is true 
that he was magnanimous, as far from envy as a 
man can be: but I think one mistakes his quality 
if one doesn't accept his judgment. I prefer to 
believe in his own statement in A Mathematician's 
Apology, at the same time so proud and so humble : 

1 1 still say to myself when I am depressed and 


find myself forced to listen to pompous and tire- 
some people, "Well, I have done one thing you 
could never have done, and that is to have 
collaborated with Littlewood and Ramanujan on 
something like equal terms."' 

In any case, his precise ranking must be left to 
the historians of mathematics (though it will be an 
almost impossible job, since so much of his best 
work was done in collaboration). There is some- 
thing else, though, at which he was clearly 
superior to Einstein or Rutherford or any other 
great genius: and that is at turning any work of 
the intellect, major or minor or sheer play, into a 
work of art. It was that gift above all, I think, 
which made him, almost without realizing it, 
purvey such intellectual delight. When A Mathe- 
matician's Apology was first published, Graham 
Greene in a review wrote that along with Henry 
James's notebooks, this was the best account of 
what it was like to be a creative artist. Thinking 
about the effect Hardy had on all those round him, 
I believe that is the clue. 

He was born, in 1877, into a modest professional 
family. His father was Bursar and Art Master at 
Cranleigh, then a minor public (English for 
private) school. His mother had been senior 
mistress at the Lincoln Training College for 
teachers. Both were gifted and mathematically 
inclined. In his case, as in that of most mathema- 


ticians, the gene pool doesn't need searching for. 
Much of his childhood, unlike Einstein's, was 
typical of a future mathematician's. He was 
demonstrating a formidably high i.q,. as soon as, 
or before, he learned to talk. At the age of two 
he was writing down numbers up to millions 
(a common sign of mathematical ability). When 
he was taken to church he amused himself by 
factorizing the numbers of the hymns: he played 
with numbers from that time on, a habit which led" 
to the touching scene at Ramanujan's sick-bed: 
the scene is well known, but later on I shall not be 
able to resist repeating it. 

It was an enlightened, cultivated, highly 
literate Victorian childhood. His parents were 
probably a little obsessive, but also very kind. 
Childhood in such a Victorian family was as 
gentle a time as anything we could provide, 
though probably intellectually somewhat more 
exacting. His was unusual in just two respects. In 
the first place, he suffered from an acute self-con- 
sciousness at an unusually early age, long before 
he was twelve. His parents knew he was pro- 
digiously clever, and so did he. He came top of his 
class in all subjects. But, as the result of coming 
top of his class, he had to go in front of the school 
to receive prizes: and that he could not bear. 
Dining with me one night, he said that he deliber- 
ately used to try to get his answers wrong so as to 


be spared this intolerable ordeal. His capacity for 
dissimulation, though, was always minimal: he 
got the prizes all the same. 

Some of this self-consciousness wore off. He 
became competitive. As he says in the Apology: 'I 
do not remember having felt, as a boy, any 
passion for mathematics, and such notions as I 
may have had of the career of a mathematician 
were far from noble. I thought of mathematics in 
terms of examinations and scholarships : I wanted 
to beat other boys, and this seemed to be the way 
in which I could do so most decisively. ' Neverthe- 
less, he had to live with an over-delicate nature. 
He seems to have been born with three skins too 
few. Unlike Einstein, who had to subjugate his 
powerful ego in the study of the external world 
before he could attain his moral stature, Hardy 
had to strengthen an ego which wasn't much 
protected. This at times in later life made him self- 
assertive (as Einstein never was) when he had to 
take a moral stand. On the other hand, it gave 
him his introspective insight and beautiful candour, 
so that he could speak of himself with absolute 
simplicity (as Einstein never could). 

I believe this contradiction, or tension, in his 
temperament was linked with a curious tic in his 
behaviour. He was the classical anti-narcissist. He 
could not endure having his photograph taken : so 
far as I know, there are only five snapshots in 


existence. He would not have any looking glass in 
his rooms, not even a shaving mirror. When he 
went to a hotel, his first action was to cover all the 
looking-glasses with towels. This would have been 
odd enough, if his face had been like a gargoyle : 
superficially it might seem odder, since all his life 
he was good-looking quite out of the ordinary. But, 
of course, narcissism and anti-narcissism have 
nothing to do with looks as outside observers see 

This behaviour seems eccentric, and indeed it 
was. Between him and Einstein, though, there 
was a difference in kind. Those who spent much 
time with Einstein— such as Infeld— found him 
grow stranger, less like themselves, the longer 
they knew him. I am certain that I should have 
felt the same. With Hardy the opposite was true. 
His behaviour was often different, bizarrely so, from 
ours: but it came to seem a kind of superstructure 
set upon a nature which wasn't all that different 
from our own, except that it was more delicate, 
less padded, finer-nerved. 

The other unusual feature of his childhood was 
more mundane: but it meant the removal of all 
practical obstacles throughout his entire career. 
Hardy, with his limpid honesty, would have been 
the last man to be finicky on this matter. He 
knew what privilege meant, and he knew that he 
had possessed it. His family had no money, only a 


schoolmaster's income, but they were in touch 
with the best educational advice of late nineteenth- 
century England. That particular kind of informa- 
tion has always been more significant in this 
country than any amount of wealth. The scholar- 
ships have been there all right, if one knew how 
to win them. There was never the slightest chance 
of the young Hardy being lost — as there was of the 
young Wells or the young Einstein. From the age 
of twelve he had only to survive, and his talents 
would be looked after. 

At twelve, in fact, he was given a scholarship at 
Winchester, then and for long afterwards the best 
mathematical school in England, simply on the 
strength of some mathematical work he had done 
at Cranleigh. (Incidentally, one wonders if any 
great school could be so elastic nowadays?) There 
he was taught mathematics in a class of one: in 
classics he was as good as the other top collegers. 
Later, he admitted that he had been well- 
educated, but he admitted it reluctantly. He dis- 
liked the school, except for its classes. Like all 
Victorian public schools, Winchester was a pretty 
rough place. He nearly died one winter. He 
envied Littlewood in his cared-for home as a day 
boy at St Paul's or other friends at our free-and- 
easy grammar schools. He never went near 
Winchester after he had left it: but he left it, with 
the inevitability of one who had got on to the 


right tramlines, with an open scholarship to 

He had one curious grievance against Win- 
chester. He was a natural ball-games player with 
a splendid eye. In his fifties he could usually beat 
the university second string at real tennis, and in 
his sixties I saw him bring off startling shots in the 
cricket nets. Yet he had not had an hour's 
coaching at Winchester : his method was defective : 
if he had been coached, he thought, he would have 
been a really good batsman, not quite first-class, 
but not too far away. Like all his judgments on 
himself, I believe that one is quite true. It is 
strange that, at the zenith of Victorian games- 
worship, such a talent was utterly missed. I suppose 
no one thought it worth looking for in the school's 
top scholar, so frail and sickly, so defensively shy. 

It would have been natural for a Wykehamist 
of his period to go to New College. That wouldn't 
have made much difference to his professional 
career (though, since he always liked Oxford 
better than Cambridge, he might have stayed 
there all his life, and some of us would have 
missed a treat). He decided to go to Trinity 
instead, for a reason that he describes, humorously 
but with his usual undecoratcd truth, in the 
Apology. 'I was about fifteen when (in a rather 
odd way) my ambitions took a sharper turn. There 
is a book by "Alan St Aubyn" (actually Mrs 


Frances Marshall) called A Fellow of Trinity, one of 
a series dealing with what is supposed to be 
Cambridge college life . . . There are two heroes, a 
primary hero called Flowers, who is almost wholly 
good, and a secondary hero, a much weaker vessel, 
called Brown. Flowers and Brown find many dan- 
gers in university life . . . Flowers survives all these 
troubles, is Second Wrangler and succeeds auto- 
matically to a Fellowship (as I suppose he would 
have done then). Brown succumbs, ruins his 
parents, takes to drink, is saved from delirium 
tremens during a thunderstorm only by the prayers 
of the Junior Dean, has much difficulty in obtain- 
ing even an Ordinary Degree, and ultimately 
becomes a missionary. The friendship is not shat- 
tered by these unhappy events, and Flowers's 
thoughts stray to Brown, with affectionate pity, as 
he drinks port and eats walnuts for the first time in 
Senior Combination Room. 

' Now Flowers was a decent enough fellow (so far 
as " Alan St Aubyn " could draw one) , but even my 
unsophisticated mind refused to accept him as 
clever. If he could do these things, why not I? In 
particular, the final scene in Combination Room 
fascinated me completely, and from that time, 
until I obtained one, mathematics meant to me 
primarily a Fellowship of Trinity. ' 

Which he duly obtained, after getting the 
highest place in the Mathematical Tripos Part II, 


at the age of 22. On the way, there were two 
minor vicissitudes. The first was theological, in the 
high Victorian manner. Hardy had decided — I 
think before he left Winchester — that he did not 
believe in God. With him, this was a black-and- 
white decision, as sharp and clear as all other 
concepts in his mind. Chapel at Trinity was com- 
pulsory. Hardy told the Dean, no doubt with his 
own kind of shy certainty, that he could not con- 
scientiously attend. The Dean, who must have 
been a jack-in-office, insisted that Hardy should 
write to his parents and tell them so. They were 
orthodox religious people, and the Dean knew, 
and Hardy knew much more, that the news would 
give them pain — pain such as we, seventy years 
later, cannot easily imagine. 

Hardy struggled with his conscience. He wasn't 
worldly enough to slip the issue. He wasn't even 
worldly enough — he told me one afternoon at 
Fenner's, for the wound still rankled — to take the 
advice of more sophisticated friends, such as 
George Trevelyan and Desmond MacCarthy, who 
would have known how to handle the matter. In 
the end he wrote the letter. Partly because of that 
incident, his religious disbelief remained open and 
active ever after. He refused to go into any college 
chapel even for formal business, like electing a 
master. He had clerical friends, but God was his 
personal enemy. In all this there was an echo of 


the nineteenth century: but one would be wrong, 
as always with Hardy, not to take him at his word. 

Still, he turned it into high jinks. I remember, 
one day in the thirties, seeing him enjoy a minor 
triumph. It happened in a Gentlemen v. Players 
match at Lord's. It was early in the morning's 
play, and the sun was shining over the pavilion. 
One of the batsmen, facing the Nursery end, com- 
plained that he was unsighted by a reflection from 
somewhere unknown. The umpires, puzzled, pad- 
ded round by the sight-screen. Motor-cars? No. 
Windows? None on that side of the ground. At 
last, with justifiable triumph, an umpire traced the 
reflection down — it came from a large pectoral 
cross reposing on the middle of an enormous 
clergyman. Politely the umpire asked him to take 
it off. Close by, Hardy was doubled up in Mephisto- 
phelian delight. That lunch time, he had no 
leisure for eating: he was writing postcards (post- 
cards and telegrams were his favourite means of 
communication) to each of his clerical friends. 

But in his war against God and God's surrogates, 
victory was not all on one side. On a quiet and 
lovely May evening at Fenner's, round about the 
same period, the chimes of six o'clock fell across 
the ground. 'It's rather unfortunate,' said Hardy 
simply, ' that some of the happiest hours of my 
life should have been spent within sound of a 
Roman Catholic church. ' 


The second minor upset of his undergraduate 
years was professional. Almost since the time of 
Newton, and all through the nineteenth century, 
Cambridge had been dominated by the examina- 
tion for the old Mathematical Tripos. The English 
have always had more faith in competitive exam- 
inations than any other people (except perhaps 
the Imperial Chinese) : they have conducted these 
examinations with traditional justice : but they have 
often shown remarkable woodenness in deciding 
what the examinations should be like. That is, 
incidentally, true to this day. It was certainly true 
of the Mathematical Tripos in its glory. It was an 
examination in which the questions were usually 
of considerable mechanical difficulty — but un- 
fortunately did not give any opportunity for the 
candidate to show mathematical imagination or 
insight or any quality that a creative mathe- 
matician needs. The top candidates (the Wranglers 
— a term which still survives, meaning a First 
Class) were arranged, on the basis of marks, in 
strict numerical order. Colleges had celebrations 
when one of their number became Senior Wran- 
gler: the first two or three Wranglers were im- 
mediately elected Fellows. 

It was all very English. It had only one dis- 
advantage, as Hardy pointed out with his polemic 
clarity, as soon as he had become an eminent 
mathematician and was engaged, together with 


his tough ally Littlewood, in getting the system 
abolished: it had effectively ruined serious mathe- 
matics in England for a hundred years. 

In his first term at Trinity, Hardy found himself 
caught in this system. He was to be trained as a 
racehorse, over a course of mathematical exercises 
which at nineteen he knew to be meaningless. He 
was sent to a famous coach, to whom most 
potential Senior Wranglers went. This coach 
knew all the obstacles, all the tricks of the exam- 
iners, and was sublimely uninterested in the 
subject itself. At this point the young Einstein 
would have rebelled: he would either have left 
Cambridge or done no formal work for the next 
three years. But Hardy was born into the more 
intensely professional English climate (which has 
its merits as well as its demerits). After considering 
changing his subject to history, he had the sense to 
find a real mathematician to teach him. Hardy 
paid him a tribute in the Apology: 'My eyes were 
first opened by Professor Love, who taught me for 
a few terms and gave me my first serious concep- 
tion of analysis. But the great debt which I owe 
to him — he was, after all, primarily an applied 
mathematician — was his advice to read Jordan's 
famous Cours a" Analyse: and I shall never forget the 
astonishment with which I read that remarkable 
work, the first inspiration for so many mathe- 
maticians of my generation, and learned for the 


first time as I read it what mathematics really 
meant. From that time onwards I was in my way 
a real mathematician, with sound mathematical 
ambitions and a genuine passion for mathematics. ' 

He was Fourth Wrangler in 1898. This faintly 
irritated him, he used to confess. He was enough of 
a natural competitor to feel that, though the race 
was ridiculous, he ought to have won it. In 1900 
he took Part II of the Tripos, a more respect- 
worthy examination, and got his right place and 
his Fellowship. 

From that time on, his life was in essence 
settled. He knew his purpose, which was to bring 
rigour into English mathematical analysis. He 
did not deviate from the researches which he 
called ' the one great permanent happiness of my 
life'. There were no anxieties about what he should 
do. Neither he nor anyone else doubted his great 
talent. He was elected to the Royal Society at 

In many senses, then, he was unusually lucky. 
He did not have to think about his career. From 
the time he was twenty-three he had all the leisure 
that a man could want, and as much money as he 
needed. A bachelor don in Trinity in the 1900's 
was comfortably off. Hardy was sensible about 
money, spent it when he felt impelled (sometimes 
for singular purposes, such as fifty-mile taxi-rides), 
and otherwise was not at all unworldly about in- 


vestments. He played his games and indulged his 
eccentricities. He was living in some of the best 
intellectual company in the world — G. E. Moore, 
Whitehead, Bertrand Russell, Trevelyan, the high 
Trinity society which was shortly to find its 
artistic complement in Bloomsbury. (Hardy him- 
self had links with Bloomsbury, both of personal 
friendship and of sympathy.) In a brilliant circle, 
he was one of the most brilliant young men — and, 
in a quiet way, one of the most irrepressible. 

I will anticipate now what I shall say later. His 
life remained the life of a brilliant young man 
until he was old: so did his spirit: his games, his 
interests, kept the lightness of a young don's. And, 
like many men who keep a young man's interests 
into their sixties, his last years were the darker for it. 

Much of his life, though, he was happier than 
most of us. He had a great many friends, of 
surprisingly different kinds. These friends had to 
pass some of his private tests: they needed to 
possess a quality which he called 'spin' (this is a 
cricket term, and untranslatable: it implies a 
certain obliquity or irony of approach : of recent 
public figures, Macmillan and Kennedy would 
get high marks for spin, Churchill and Eisenhower 
not). But he was tolerant, loyal, extremely high- 
spirited, and in an undemonstrative way fond 
of his friends. I once was compelled to go and see 
him in the morning, which was always his set 


time for mathematical work. He was sitting at his 
desk, writing in his beautiful calligraphy. I mur- 
mured some commonplace politeness about hoping 
that I wasn't disturbing him. He suddenly dis- 
solved into his mischievous grin. 'As you ought to 
be able to notice, the answer to that is that you 
are. Still, I'm usually glad to see you.' In the 
sixteen years we knew each other, he didn't say 
anything more demonstrative than that: except on 
his deathbed, when he told me that he looked 
forward to my visits. 

I think my experience was shared by most of his 
close friends. But he had, scattered through his 
life, two or three other relationships, different in 
kind. These were intense affections, absorbing, 
non-physical but exalted. The one I knew about 
was for a young man whose nature was as spirit- 
ually delicate as his own. I believe, though I only 
picked this up from chance remarks, that the same 
was true of the others. To many people of my 
generation, such relationships would seem either 
unsatisfactory or impossible. They were neither 
the one nor the other; and, unless one takes them 
for granted, one doesn't begin to understand the 
temperament of men like Hardy (they are rare, 
but not as rare as white rhinoceroses), nor the 
Cambridge society of his time. He didn't get the 
satisfactions that most of us can't help finding: but 
he knew himself unusually well, and that didn't 


make him unhappy. His inner life was his own, 
and very rich. The sadness came at the end. Apart 
from his devoted sister, he was left with no one 
close to him. 

With sardonic stoicism he says in the Apology — 
which for all its high spirits is a book of desperate 
sadness — that when a creative man has lost the 
power or desire to create — 'It is a pity but in that 
case he does not matter a great deal anyway, and 
it would be silly to bother about him. ' That is how 
he treated his personal life outside mathematics. 
Mathematics was his justification. It was easy to 
forget this, in the sparkle of his company: just as it 
was easy in the presence of Einstein's moral 
passion to forget that to himself his justification 
was his search for the physical laws. Neither of 
those two ever forgot it. This was the core of their 
lives, from young manhood to death. 

Hardy, unlike Einstein, did not make a quick 
start. His early papers, between 1900 and 191 1, 
were good enough to get him into the Royal 
Society and win him an international reputation: 
but he did not regard them as important. Again, 
this wasn't false modesty: it was the judgment of 
a master who knew to an inch which of his work 
had value and which hadn't. 

In 191 1 he began a collaboration with Little- 
wood which lasted thirty-five years. In 191 3 
he discovered Ramanujan and began another 


collaboration. All his major work was done in these 
two partnerships, most of it in the one with 
Littlewood, the most famous collaboration in the 
history of mathematics. There has been nothing 
like it in any science, or, so far as I know, in any 
other field of creative activity. Together they 
produced nearly a hundred papers, a good many 
of them 'in the Bradman class'. Mathematicians 
not intimate with Hardy in his later years, nor 
with cricket, keep repeating that his highest term 
of praise was 'in the Hobbs class'. It wasn't: very 
reluctantly, because Hobbs was one of his pets, he 
had to alter the order of merit. I once had a 
postcard from him, probably in 1938, saying 
'Bradman is a whole class above any batsman 
who has ever lived : if Archimedes, Newton and 
Gauss remain in the Hobbs class, I have to admit 
the possibility of a class above them, which I find 
difficult to imagine. They had better be moved 
from now on into the Bradman class. ' 

The Hardy-Littlewood researches dominated 
English pure mathematics, and much of world 
pure mathematics, for a generation. It is too early 
to say, so mathematicians tell me, to what extent 
they altered the course of mathematical analysis : 
nor how influential their work will appear in a 
hundred years. Of its enduring value there is no 

Theirs was, as I have said, the greatest of all 


collaborations. But no one knows how they did it: 
unless Littlewood tells us, no one will ever know. 
I have already given Hardy's judgment that 
Littlewood was the more powerful mathematician 
of the two : Hardy once wrote that he knew of ' no 
one else who could command such a combination 
of insight, technique and power'. Littlewood was 
and is a more normal man than Hardy, just as 
interesting and probably more complex. He never 
had Hardy's taste for a kind of refined intellectual 
flamboyance, and so was less in the centre of the 
academic scene. This led to jokes from European 
mathematicians, such as that Hardy had invented 
him so as to take the blame in case there turned 
out anything wrong with one of their theorems. 
In fact, he is a man of at least as obstinate an 
individuality as Hardy himself. 

At first glance, neither of them would have 
seemed the easiest of partners. It is hard to 
imagine either of them suggesting the collabora- 
tion in the first place. Yet one of them must have 
done. No one has any evidence how they set about 
it. Through their most productive period, they 
were not at the same university. Harald Bohr 
(brother of Niels Bohr, and himself a fine mathe- 
matician) is reported as saying that one of their 
principles was this : if one wrote a letter to the other, 
the recipient was under no obligation to reply to 
it, or even to read it. 


I can't contribute anything. Hardy talked to me, 
over a period of many years, on almost every 
conceivable subject, except the collaboration. He 
said, of course, that it had been the major fortune 
of his creative career: he spoke of Littlewood in the 
terms I have given: but he never gave a hint of 
their procedures. I didn't know enough mathe- 
matics to understand their papers, but I picked up 
some of their language. If he had let slip anything 
about their methods, I don't think I should have 
missed it. I am fairly certain that the secrecy — 
quite uncharacteristic of him in matters which to 
most would seem more intimate — was deliberate. 

About his discovery of Ramanujan, he showed 
no secrecy at all. It was, he wrote, the one roman- 
tic incident in his life : anyway, it is an admirable 
story, and one which showers credit on nearly 
everyone (with two exceptions) in it. One morning 
early in 191 3, he found, among the letters on his 
breakfast table, a large untidy envelope decorated 
with Indian stamps. When he opened it, he found 
sheets of paper by no means fresh, on which, in a 
non-English holograph, were line after line of 
symbols. Hardy glanced at them without enthu- 
siasm. He was by this time, at the age of thirty-six, 
a world famous mathematician : and world famous 
mathematicians, he had already discovered, are 
unusually exposed to cranks. He was accustomed 
to receiving manuscripts from strangers, proving 


the prophetic wisdom of the Great Pyramid, the 
revelations of the Elders of Zion, or the crypto- 
grams that Bacon had inserted in the plays of the 
so-called Shakespeare. 

So Hardy felt, more than anything, bored. He 
glanced at the letter, written in halting English, 
signed by an unknown Indian, asking him to give 
an opinion of these mathematical discoveries. The 
script appeared to consist of theorems, most of 
them wild or fantastic looking, one or two already 
well-known, laid out as though they were original. 
There were no proofs of any kind. Hardy was not 
only bored, but irritated. It seemed like a curious 
kind of fraud. He put the manuscript aside, and 
went on with his day's routine. Since that routine 
did not vary throughout his life, it is possible to 
reconstruct it. First he read The Times over his 
breakfast. This happened in January, and if there 
were any Australian cricket scores, he would start 
with them, studied with clarity and intense 

Maynard Keynes, who began his career as a 
mathematician and who was a friend of Hardy's, 
once scolded him : if he had read the stock exchange 
quotations half an hour each day with the same 
concentration he brought to the cricket scores, he 
could not have helped becoming a rich man. 

Then, from about nine to one, unless he was 
giving a lecture, he worked at his own mathe- 


matics. Four hours creative work a day is about 
the limit for a mathematician, he used to say. 
Lunch, a light meal, in hall. After lunch he loped 
off for a game of real tennis in the university 
court. (If it had been summer, he would have 
walked down to Fenner's to watch cricket.) In the 
late afternoon, a stroll back to his rooms. That 
particular day, though, while the timetable wasn't 
altered, internally things were not going according 
to plan. At the back of his mind, getting in the 
way of his complete pleasure in his game, the 
Indian manuscript nagged away. Wild theorems. 
Theorems such as he had never seen before, nor 
imagined. A fraud of genius? A question was 
forming itself in his mind. As it was Hardy's mind, 
the question was forming itself with epigrammatic 
clarity: is a fraud of genius more probable than an 
unknown mathematician of genius? Clearly the 
answer was no. Back in his rooms in Trinity, he 
had another look at the script. He sent word to 
Little wood (probably by messenger, certainly not 
by telephone, for which, like all mechanical con- 
trivances including fountain pens, he had a deep 
distrust) that they must have a discussion after hall. 
When the meal was over, there may have been a 
slight delay. Hardy liked a glass of wine, but, 
despite the glorious vistas of 'Alan St. Aubyn' 
which had fired his youthful imagination, he 
found he did not really enjoy lingering in the 


combination-room over port and walnuts. Little- 
wood, a good deal more homme moyen sensuel, did. 
So there may have been a delay. Anyway, by nine 
o'clock or so they were in one of Hardy's rooms, 
with the manuscript stretched out in front of them. 

That is an occasion at which one would have 
liked to be present. Hardy, with his combination 
of remorseless clarity and intellectual panache 
(he was very English, but in argument he showed 
the characteristics that Latin minds have often 
assumed to be their own) : Littlewood, imaginative, 
powerful, humorous. Apparently it did not take 
them long. Before midnight they knew, and knew 
for certain. The writer of these manuscripts was 
a man of genius. That was as much as they could 
judge, that night. It was only later that Hardy 
decided that Ramanujan was, in terms of natural 
mathematical genius, in the class of Gauss and 
Euler: but that he could not expect, because of 
the defects of his education, and because he had 
come on the scene too late in the line of mathe- 
matical history, to make a contribution on the 
same scale. 

It all sounds easy, the kind of judgment great 
mathematicians should have been able to make. 
But I mentioned that there were two persons who 
do not come out of the story with credit. Out of 
chivalry Hardy concealed this in all that he said 
or wrote about Ramanujan. The two people con- 


cerned have now been dead, however, for many 
years, and it is time to tell the truth. It is simple. 
Hardy was not the first eminent mathematician 
to be sent the Ramanujan manuscripts. There had 
been two before him, both English, both of the 
highest professional standard. They had each 
returned the manuscripts without comment. I 
don't think history relates what they said, if any- 
thing, when Ramanujan became famous. Anyone 
who has been sent unsolicited material will have a 
sneaking sympathy with them. 

Anyway, the following day Hardy went into 
action. Ramanujan must be brought to England, 
he decided. Money was not a major problem. 
Trinity has usually been good at supporting un- 
orthodox talent (the college did the same for 
Kapitsa a few years later). Once Hardy was de- 
termined, no human agency could have stopped 
Ramanujan, but they needed a certain amount 
of help from a superhuman one. 

Ramanujan turned out to be a poor clerk in 
Madras, living with his wife on twenty pounds a 
year. But he was also a Brahmin, unusually strict 
about his religious observances, with a mother 
who was even stricter. It seemed impossible that 
he could break the proscriptions and cross the 
water. Fortunately his mother had the highest 
respect for the goddess of Namakkal. One morning 
Ramanujan's mother made a startling announcc- 


ment. She had had a dream on the previous night, 
in which she saw her son seated in a big hall 
among a group of Europeans, and the goddess of 
Namakkal had commanded her not to stand in 
the way of her son fulfilling his life's purpose. This, 
say Ramanujan's Indian biographers, was a very 
agreeable surprise to all concerned. 

In 1 91 4 Ramanujan arrived in England. So far 
as Hardy could detect (though in this respect I 
should not trust his insight far) Ramanujan, despite 
the difficulties of breaking the caste proscriptions, 
did not believe much in theological doctrine, 
except for a vague pantheistic benevolence, any 
more than Hardy did himself. But he did certainly 
believe in ritual. When Trinity put him up in 
college — within four years he became a Fellow — 
there was no 'Alan St. Aubyn' apolausticity for 
him at all. Hardy used to find him ritually changed 
into his pyjamas, cooking vegetables rather 
miserably in a frying pan in his own room. 

Their association was a strangely touching one. 
Hardy did not forget that he was in the presence 
of genius : but genius that was, even in mathematics, 
almost untrained. Ramanujan had not been able 
to enter Madras University because he could not 
matriculate in English. According to Hardy's 
report, he was always amiable and good-natured, 
but no doubt he sometimes found Hardy's con- 
versation outside mathematics more than a little 


baffling. He seems to have listened with a patient 
smile on his good, friendly, homely face. Even 
inside mathematics they had to come to terms 
with the difference in their education. Ramanujan 
was self-taught : he knew nothing of the modern 
rigour: in a sense he didn't know what a proof was. 
In an uncharacteristically sloppy moment, Hardy 
once wrote that if he had been better educated, he 
would have been less Ramanujan. Coming back 
to his ironic senses, Hardy later corrected himself 
and said that the statement was nonsense. If 
Ramanujan had been better educated, he would 
have been even more wonderful than he was. In 
fact, Hardy was obliged to teach him some formal 
mathematics as though Ramanujan had been a 
scholarship candidate at Winchester. Hardy said 
that this was the most singular experience of his 
life: what did modern mathematics look like to 
someone who had the deepest insight, but who had 
literally never heard of most of it? 

Anyway, they produced together five papers of 
the highest class, in which Hardy showed supreme 
originality of his own (more is known of the details 
of this collaboration than of the Hardy-Littlewood 
one). Generosity and imagination were, for once, 
rewarded in full. 

This is a story of human virtue. Once people had 
started behaving well, they went on behaving 
better. It is good to remember that England gave 


Ramanujan such honours as were possible. The 
Royal Society elected him a Fellow at the age 
of thirty (which, even for a mathematician, is very 
young). Trinity also elected him a Fellow in the 
same year. He was the first Indian to be given 
either of these distinctions. He was amiably grate- 
ful. But he soon became ill. It was difficult, in 
war-time, to move him to a kinder climate. 

Hardy used to visit him, as he lay dying in 
hospital at Putney. It was on one of those visits 
that there happened the incident of the taxi-cab 
number. Hardy had gone out to Putney by taxi, as 
usual his chosen method of conveyance. He went 
into the room where Ramanujan was lying. Hardy, 
always inept about introducing a conversation, 
said, probably without a greeting, and certainly as 
his first remark: 'I thought the number of my taxi- 
cab was 1729. It seemed to me rather a dull 
number.' To which Ramanujan replied: 'No, 
Hardy ! No, Hardy ! It is a very interesting number. 
It is the smallest number expressible as the sum of 
two cubes in two different ways. ' 

That is the exchange as Hardy recorded it. It 
must be substantially accurate. He was the most 
honest of men ; and further, no one could possibly 
have invented it. 

Ramanujan died of tuberculosis, back in Madras, 
two years after the war. As Hardy wrote in the 
Apology, in his roll-call of mathematicians : c Galois 


died at twenty-one, Abel at twenty-seven, Rama- 
nujan at thirty-three, Riemann at forty. . . I do not 
know an instance of a major mathematical advance 
initiated by a man past fifty.' 

If it had not been for the Ramanujan collabora- 
tion, the 1 914-18 war would have been darker for 
Hardy than it was. But it was dark enough. It left 
a wound which reopened in the second war. He 
was a man of radical opinions all his life. His 
radicalism, though, was tinged with the turn-of- 
the-century enlightenment. To people of my 
generation, it sometimes seemed to breathe a 
lighter, more innocent air, than the one we knew. 

Like many of his Edwardian intellectual friends, 
he had a strong feeling for Germany. Germany had, 
after all, been the great educating force of the 
nineteenth century. To Eastern Europe, to Russia, 
to the United States, it was the German universities 
which had taught the meaning of research. Hardy 
hadn't much use for German philosophy or 
German literature: his tastes were too classical 
for that. But in most respects the German culture, 
including its social welfare, appeared to him 
higher than his own. 

Unlike Einstein, who had a much more rugged 
sense of political existence, Hardy did not know 
much of Wilhelmine Germany at first hand. And, 
though he was the least vain of people, he would 
have been less than human if he had not enjoyed 


being more appreciated in Germany than in his 
own country. There is a pleasant anecdote, 
dating from this period, in winch Hilbert, one of 
the greatest of German mathematicians, heard 
that Hardy lived in a not specially agreeable set of 
rooms in Trinity (actually in Whewell's Court). 
Hilbert promptly wrote in measured terms to the 
Master, pointing out that Hardy was the best 
mathematician, not only in Trinity but in England, 
and should therefore have the best rooms. 

So Hardy, like Russell and many of the high 
Cambridge intelligentsia, did not believe that the 
war should have been fought. Further, with his 
ingrained distrust of English politicians, he thought 
the balance of wrong was on the English side. He 
could not find a satisfactory basis for conscientious 
objection; his intellectual rigour was too strong 
for that. In fact, he volunteered for service under 
the Derby scheme, and was rejected on medical 
grounds. But he felt increasingly isolated in 
Trinity, much of which was vociferously bellicose. 

Russell was dismissed from his lectureship, in 
circumstances of overheated complexity (Hardy 
was to write the only detailed account of the case a 
quarter of a century later, in order to comfort 
himself in another war). Hardy's close friends 
were away at the war. Littlewood was doing bal- 
listics as a Second Lieutenant in the Royal Artil- 
lery. Owing to his cheerful indifference he had the 


distinction of remaining a Second Lieutenant 
through the four years of war. Their collaboration 
was interfered with, though not entirely stopped. 
It was the work of Ramanujan which was Hardy's 
solace during the bitter college quarrels. 

I sometimes thought he was, for once, less than 
fair to his colleagues. Some were pretty crazed, as 
men are in war-time. But some were long- 
suffering and tried to keep social manners going. 
After all, it was a triumph of academic uprightness 
that they should have elected his protege" Ramanu- 
jan, at a time when Hardy was only just on 
speaking terms with some of the electors, and not 
at all with others. 

Still, he was harshly unhappy. As soon as he 
could, he left Cambridge. He was offered a chair at 
Oxford in 191 9: and immediately walked into the 
happiest time of his life. He had already done 
great work with Ramanujan and Littlewood, but 
now the collaboration with Littlewood rose to its 
full power. Hardy was, in Newton's phrase, 'in 
the prime of his life for invention', and this came 
in his early forties, unusually late for a mathe- 

Coming so late, this creative surge gave him the 
feeling, more important to him than to most men, 
of timeless youth. He was living the young man's life 
which was first nature to him. He played more 
real tennis, and got steadily better at it (real tennis 


was an expensive game and took a largish slice out 
of a professorial income) . He made a good many 
visits to American universities, and loved the 
country. He was one of the few Englishmen of his 
time who was fond, to an extent approximately 
equal, of the United States and the Soviet Union. 
He was certainly the only Englishman of his or 
any other time who wrote a serious suggestion to 
the Baseball Commissioners, proposing a technical 
emendation to one of the rules. The twenties, for 
him and for most liberals of his generation, was a 
false dawn. He thought the misery of the war was 
swept away into the past. 

He was at home in New College as he had 
never been in Cambridge. The warm domestic 
conversational Oxford climate was good for him. 
It was there, in a college at that time small and 
intimate, that he perfected his own style of con- 
versation. There was company eager to listen to 
him after hall. They could take his eccentricities. 
He was not only a great and good man, they 
realized, but an entertaining one. If he wanted to 
play conversational games, or real (though bizarre) 
games on the cricket field, they were ready to act 
as foils. In a casual and human fashion, they made 
a fuss of him. He had been admired and esteemed 
before, but not made a fuss of in that fashion. 

No one seemed to care — it was a gossipy college 
joke — that he had a large photograph of Lenin 


in his rooms. Hardy's radicalism was somewhat 
unorganized, but it was real. He had been born, as 
I have explained, into a professional family: al- 
most all his life was spent among the haute 
bourgeoisie: but in fact he behaved much more 
like an aristocrat, or more exactly like one of the 
romantic projections of an aristocrat. Some of this 
attitude, perhaps, he had picked up from his 
friend Bertrand Russell. But most of it was innate. 
Underneath his shyness, he just didn't give a damn. 
He got on easily, without any patronage, with 
the poor, the unlucky and diffident, those who 
were handicapped by race (it was a symbolic, 
stroke of fate that he discovered Ramanujan). He 
preferred them to the people whom he called 
the large bottomed: the description was more 
psychological than physiological, though there was 
a famous nineteenth-century Trinity aphorism by 
Adam Sedgwick: 'No one ever made a success in 
this world without a large bottom.' To Hardy the 
large bottomed were the confident, booming, im- 
perialist bourgeois English. The designation in- 
cluded most bishops, headmasters, judges, and all 
politicians, with the single exception of Lloyd 

Just to show his allegiances, he accepted one 
public office. For two years (1924-26) he was 
President of the Association of Scientific Workers. 
He said sarcastically that he was an odd choice, 


being ' the most unpractical member of the most 
unpractical profession in the world'. But in im- 
portant things he was not so unpractical. He was 
deliberately standing up to be counted. When, 
much later, I came to work with Frank Cousins, it 
gave me a certain quiet pleasure to reflect that I 
had had exactly two friends who had held office in 
the Trade Union movement, him and G. H. 

That late, not quite Indian, summer in Oxford 
in the twenties was so happy that people wondered 
that he ever returned to Cambridge. Which he 
did in 1 93 1 . I think there were two reasons. First 
and most decisive, he was a great professional. 
Cambridge was still the centre of English mathe- 
matics, and the senior mathematical chair there 
was the correct place for a professional. Secondly, 
and rather oddly, he was thinking about his old 
age. Oxford colleges, in many ways so human and 
warm, are ruthless with the old: if he remained at 
New College he would be turned out of his rooms 
as soon as he retired, at the age of sixty-five, from 
his professorship. Whereas, if he returned to 
Trinity, he could stay in college until he died. 
That is in effect what he managed to do. 

When he came back to Cambridge — which was 
the time that I began to know him — he was in the 
afterglow of his great period. He was still happy. 
He was still creative, not so much as in the 


twenties, but enough to make him feel that the 
power was still there. He was as spirited as he had 
been at New College. So we had the luck to see 
him very nearly at his best. 

In the winters, after we had become friendly, 
we gave each other dinner in our respective 
colleges once a fortnight. When summer came, it 
was taken for granted that we should meet at the 
cricket ground. Except on special occasions he 
still did mathematics in the morning, and did not 
arrive at Fenner's until after lunch. He used to 
walk round the cinderpath with a long, loping, 
dumping-footed stride (he was a slight spare man, 
physically active even in his late fifties, still 
playing real tennis) head down, hair, tie, sweaters, 
papers all flowing , a figure that caught everyone's 
eyes. 'There goes a Greek poet, I'll be bound,' 
once said some cheerful farmer as Hardy passed 
the score-board. He made for his favourite place, 
opposite the pavilion, where he could catch each 
ray of sun — he was obsessively heliotropic. In 
order to deceive the sun into shining, he brought 
with him, even on a fine May afternoon, what he 
called his 'anti-God battery'. This consisted of 
three or four sweaters, an umbrella belonging to 
his sister, and a large envelope containing mathe- 
matical manuscripts, such as a Ph.D. dissertation, 
a paper which he was refereeing for the Royal 
Society, or some tripos answers. He would explain 


to an acquaintance that God, believing that Hardy 
expected the weather to change and give him a 
chance to work, counter-suggestibly arranged that 
the sky should remain cloudless. 

There he sat. To complete his pleasure in a long 
afternoon watching cricket, he liked the sun to be 
shining and a companion to join in the fun. 
Technique, tactics, formal beauty — those were 
the deepest attractions of the game for him. I 
won't try to explain them: they are incommu- 
nicable unless one knows the language: just as 
some of Hardy's classical aphorisms are inexplic- 
able unless one knows cither the language of 
cricket or of the theory of numbers, and preferably 
both. Fortunately for a good many of our friends, 
he also had a relish for the human comedy. 

He would have been the first to disclaim that he 
had any special psychological insight. But he was 
the most intelligent of men, he had lived with his 
eyes open and read a lot, and he had obtained a 
good generalized sense of human nature — robust, 
indulgent, satirical, and utterly free from moral 
vanity. He was spiritually candid as few men are 
(I doubt if anyone could be more candid), and he 
had a mocking horror of pretentiousness, self- 
righteous indignation, and the whole stately 
pantechnicon of the hypocritical virtues. Now 
cricket, the most beautiful of games, is also the 
most hypocritical. It is supposed to be the 


ultimate expression of the team spirit. One ought 
to prefer to make o and see one's side win, than 
make ioo and see it lose (one very great player, 
like Hardy a man of innocent candour, once 
remarked mildly that he never managed to feel 
so). This particular ethos inspired Hardy's sense 
of the ridiculous. In reply he used to expound a 
counterbalancing series of maxims. Examples : 

' Cricket is the only game where you are playing 

against eleven of the other side and ten of your own. ' 

' If you are nervous when you go in first, nothing 

restores your confidence so much as seeing the 

other man get out.' 

If his listeners were lucky, they would hear other 
remarks, not relevant to cricket, as sharp-edged in 
conversation as in his writing. In the Apology 
there are some typical specimens: here are a few 

'It is never worth a first class man's time to 
express a majority opinion. By definition, there 
are plenty of others to do that. ' 

'When I was an undergraduate one might, if 
one were sufficiently unorthodox, suggest that 
Tolstoi came within touching distance of George 
Meredith as a novelist: but, of course, no one else 
possibly could. ' (This was said about the intoxica- 
tions of fashion : it is worth remembering that he 
had lived in one of the most brilliant of all 
Cambridge generations.) 


'For any serious purpose, intelligence is a very 
minor gift. ' 

'Young men ought to be conceited: but they 
oughtn't to be imbecile.' (Said after someone had 
tried to persuade him that Finnegans Wake was 
the final literary masterpiece.) 

'Sometimes one has to say difficult things, but 
one ought to say them as simply as one knows how. ' 

Occasionally, as he watched cricket, his ball-by- 
ball interest flagged. Then he demanded that we 
should pick teams : teams of humbugs, club-men, 
bogus poets, bores, teams whose names began with 
HA (numbers one and two Hadrian and Hannibal) , 
SN, all-time teams of Trinity, Christ's, and so on. 
In these exercises I was at a disadvantage: let 
anyone try to produce a team of world-figures 
whose names start with SN. The Trinity team is 
overwhelmingly strong (Clerk Maxwell, Byron, 
Thackeray, Tennyson aren't certain of places) : 
while Christ's, beginning strongly with Milton 
and Darwin, has nothing much to show from 
number 3 down. 

Or he had another favourite entertainment. 
'Mark that man we met last night', he said, and 
someone had to be marked out of 100 in each of 
the categories Hardy had long since invented and 
defined. Stark, Bleak ('a stark man is not 
necessarily bleak: but all bleak men without 
exception want to be considered stark'), Dim, 


Old Brandy, Spin, and some others. Stark, 
Bleak and Dim are self-explanatory (the Duke 
of Wellington would get a flat ioo for Stark and 
Bleak, and o for Dim). Old Brandy was 
derived from a mythical character who said that 
he never drank anything but old brandy. Hence, 
by extrapolation, Old Brandy came to mean a 
taste that was eccentric, esoteric, but just within 
the bounds of reason. As a character (and in 
Hardy's view, though not mine, as a writer also) 
Proust got high marks for Old Brandy: so did 
F. A. Lindemann (later Lord Cherwell). 

The summer days passed. After one of the short 
Cambridge seasons, there was the University match. 
Arranging to meet him in London was not always 
simple, for, as I mentioned before, he had a morbid 
suspicion of mechanical gadgets (he never used a 
watch), in particular of the telephone. In his rooms 
in Trinity or his flat in St George's Square, he used 
to say, in a disapproving and slightly sinister tone : 
' If you fancy yourself at the telephone, there is one in 
the next room. ' Once in an emergency he had to 
ring me up : angrily his voice came at me : * I shan't 
hear a word you say, so when I'm finished I shall 
immediately put the receiver down. It's important 
you should come round between nine and ten 
tonight.' Clonk. 

Yet, punctually, he arrived at the University 
match. There he was at his most sparkling, year 


after year. Surrounded by friends, men and 
women, he was quite released from shyness. He 
was the centre of all our attention, which he 
didn't dislike. Sometimes one could hear the 
party's laughter from a quarter of the way round 
the ground. 

In those last of his happy years, everything he 
did was light with grace, order, a sense of style. 
Cricket is a game of grace and order, which is 
why he found formal beauty in it. His mathe- 
matics, so I am told, had these same aesthetic 
qualities, right up to his last creative work. I have 
given the impression, I fancy, that in private he 
was a conversational performer. To an extent, 
that was true : but he was also, on what he would 
have called c non-trivial ' occasions (meaning occa- 
sions important to either participant) a serious 
and concentrated listener. Of other eminent 
persons whom , by various chances, I knew at 
the same period, Wells was, on the whole, a worse 
listener than one expected: Rutherford distinctly 
better : Lloyd George one of the best listeners of all 
time. Hardy didn't suck impressions and know- 
ledge out of others' words, as Lloyd George did, 
but his mind was at one's disposal. When, years 
before I wrote it, he heard of the concept of 
The Masters, he cross-examined me, so that I did 
most of the talking. He produced some good 
ideas. I wish he had been able to read the book, 


which I think that he might have liked. Anyway, 
in that hope I dedicated it to his memory. 

In the note at the end of the Apology he refers to 
other discussions. One of these was long-drawn- 
out, and sometimes, on both sides, angry. On the 
second world war we each had passionate but, as 
I shall have to say a little later, different opinions. 
I didn't shift his by one inch. Nevertheless, though 
we were separated by a gulf of emotion, on the 
plane of reason he recognized what I was saying. 
That was true in any argument I had with him. 

Through the thirties he lived his own version of 
a young man's life. Then suddenly it broke. In 
1939 he had a coronary thrombosis. He recovered, 
but real tennis, squash, the physical activities he 
loved, were over for good. The war darkened him 
still further, just as the first war had. To him they 
were connected pieces of lunacy, we were all at 
fault, he couldn't identify himself with the war — 
once it was clear the country would survive — any 
more than he had done in 19 14. One of his 
closest friends died tragically. And — I think there is 
no doubt these griefs were inter-connected — his 
creative powers as a mathematician at last, in his 
sixties, left him. 

That is why A Mathematician 's Apology is, if read 
with the textual attention it deserves, a book of 
haunting sadness. Yes, it is witty and sharp with 
intellectual high spirits: yes, the crystalline clarity 


and candour are still there: yes, it is the testament 
of a creative artist. But it is also, in an understated 
stoical fashion, a passionate lament for creative 
powers that used to be and that will never come 
again. I know nothing like it in the language: 
partly because most people with the literary gift 
to express such a lament don't come to feel it: it is 
very rare for a writer to realize, with the finality 
of truth, that he is absolutely finished. 

Seeing him in those years, I couldn't help 
thinking of the price he was paying for his young 
man's life. It was like seeing a great athlete, for 
years in the pride of his youth and skill, so much 
younger and more joyful than the rest of us, 
suddenly have to accept that the gift has gone. It 
is common to meet great athletes who have gone, 
as they call it, over the hill: fairly quickly the feet 
get heavier (often the eyes last longer), the strokes 
won't come off, Wimbledon is a place to be 
dreaded, the crowds go to watch someone else. 
That is the point at which a good many athletes 
take to drink. Hardy didn't take to drink: but he 
took to something like despair. He recovered 
enough physically to have ten minutes batting at 
the nets, or to play his pleasing elaboration (with a 
complicated set of bisques) of Trinity bowls. But it 
was often hard to rouse his interest — three or four 
years before his interest in everything was so 
sparkling as sometimes to tire us all out. 'No one 
should ever be bored', had been one of his 


axioms. 'One can be horrified, or disgusted, but 
one can't be bored. ' Yet now he was often just 
that, plain bored. 

It was for that reason that some of his friends, 
including me, encouraged him to write the story of 
Bertrand Russell and Trinity in the 1914-18 war. 
People who didn't know how depressed Hardy was 
thought the whole episode was now long over and 
ought not to be resurrected. The truth was, it 
enlivened him to have any kind of purpose. The 
book was privately circulated. It has never been 
obtainable by the public, which is a pity, for it is a 
small-scale addition to academic history. 

I used such persuasion as I had to get him to 
write another book, which he had in happier days 
promised me to do. It was to be called A Day at the 
Oval and was to consist of himself watching 
cricket for a whole day, spreading himself in dis- 
quisitions on the game, human nature, his 
reminiscences, life in general. It would have been 
an eccentric minor classic: but it was never 

I wasn't much help to him in those last years. I 
was deeply involved in war-time Whitehall, I was 
preoccupied and often tired, it was an effort to get 
to Cambridge. But I ought to have made the effort 
more often than I did. I have to admit, with 
remorse, that there was, not exactly a chill, but a 
gap in sympathy between us. He lent me his flat in 


Pimlico — a dark and seedy flat with the St 
George's Square gardens outside and what he 
called an 'old brandy' attractiveness — for the 
whole of the war. But he didn't like me being so 
totally committed. People he approved of oughtn't 
to give themselves whole-heartedly to military 
functions. He never asked me about my work. He 
didn't want to talk about the war. While I, for 
my part, was impatient and didn't show anything 
like enough consideration. After all, I thought, I 
wasn't doing this job for fun : as I had to do it, I 
might as well extract the maximum interest. But 
that is no excuse. 

At the end of the war I did not return to 
Cambridge. I visited him several times in 1946. 
His depression had not lifted, he was physically 
failing, short of breath after a few yards walk. The 
long cheerful stroll across Parker's Piece, after the 
close of play, was gone for ever: I had to take him 
home to Trinity in a taxi. He was glad that I had 
gone back to writing books: the creative life was 
the only one for a serious man. As for himself, he 
wished that he could live the creative life again, no 
better than it had been before : his own life was 


I am not quoting his exact words. This was so 
unlike him that I wanted to forget, and I tried, by 
a kind of irony, to smear over what had just been 
said. So that I have never remembered precisely. 


I attempted to dismiss it to myself as a rhetorical 

In the early summer of 1947 I was sitting at 
breakfast when the telephone rang. It was Hardy's 
sister: he was seriously ill, would I come up to 
Cambridge at once, would I call at Trinity first? 
At the time I didn't grasp the meaning of the 
second request. But I obeyed it, and in the porter's 
lodge at Trinity that morning found a note from 
her: I was to go to Donald Robertson's rooms, he 
would be waiting for me. 

Donald Robertson was the Professor of Greek, 
and an intimate friend of Hardy's: he was 
another member of the same high, liberal, graceful 
Edwardian Cambridge. Incidentally, he was one 
of the few people who called Hardy by his 
Christian name. He greeted me quietly. Outside 
the windows of his room it was a calm and sunny 
morning. He said: 

'You ought to know that Harold has tried to 
kill himself.' 

Yes, he was out of danger: he was for the time 
being all right, if that was the phrase to use. But 
Donald was, in a less pointed fashion, as direct as 
Hardy himself. It was a pity the attempt had 
failed. Hardy's health had got worse : he could not 
in any case live long: even walking from his rooms 
to hall had become a strain. He had made a 
completely deliberate choice. Life on those terms 


he would not endure: there was nothing in it. He 
had collected enough barbiturates : he had tried to 
do a thorough job, and had taken too many. 

I was fond of Donald Robertson, but I had met 
him only at parties and at Trinity high table. This 
was the first occasion on which we had talked 
intimately. He said, with gentle firmness, that I 
ought to come up to see Hardy as often as I could : 
it would be hard to take, but it was an obligation : 
probably it would not be for long. We were both 
wretched. I said goodbye, and never saw him 

In the Evelyn nursing home, Hardy was lying 
in bed. As a touch of farce, he had a black eye. 
Vomiting from the drugs, he had hit his head on 
the lavatory basin. He was self-mocking. He had 
made a mess of it. Had anyone ever made a 
bigger mess? I had to enter into the sarcastic 
game. I had never felt less like sarcasm, but I had 
to play. I talked about other distinguished failures 
at bringing off suicide. What about German 
generals in the last war? Beck, Stulpnagel, they 
had been remarkably incompetent at it. It was 
bizarre to hear myself saying these things. Curiously 
enough, it seemed to cheer him up. 

After that, I went to Cambridge at least once a 
week. I dreaded each visit, but early on he said 
that he looked forward to seeing me. He talked a 
little, nearly every time I saw him, about death. 


He wanted it: he didn't fear it: what was there to 
fear in nothingness? His hard intellectual stoicism 
had come back. He would not try to kill himself 
again. He wasn't good at it. He was prepared to 
wait. With an inconsistency which might have 
pained him — for he, like most of his circle, 
believed in the rational to an extent that I thought 
irrational — he showed an intense hypochondriac 
curiosity about his own symptoms. Constantly he 
was studying the oedema of his ankles: was it 
greater or less that day? 

Mostly, though — about fifty-five minutes in 
each hour I was with him — I had to talk cricket. 
It was his only solace. I had to pretend a devotion 
to the game which I no longer felt, which in fact 
had been lukewarm in the thirties except for the 
pleasure of his company. Now I had to study the 
cricket scores as intendy as when I was a schoolboy. 
He couldn't read for himself, but he would have 
known if I was bluffing. Sometimes, for a few 
minutes, his old vivacity would light up. But if I 
couldn't think of another question or piece of news, 
he would lie there, in the kind of dark loneliness 
that comes to some people before they die. 

Once or twice I tried to rouse him. Wouldn't it 
be worth while, even if it was a risk, to go and see 
one more cricket match together? I was now better 
off than I used to be, I said. I was prepared to 
stand him a taxi, his old familiar means of trans- 


port, to any cricket ground he liked to name. At 
that he brightened. He said that I might have a 
dead man on my hands. I replied that I was ready 
to cope. I thought that he might come: he knew, 
I knew, that his death could only be a matter of 
months: I wanted to see him have one afternoon 
of something like gaiety. The next time I visited 
him he shook his head in sadness and anger. No, 
he couldn't even try: there was no point in trying. 

It was hard enough for me to have to talk 
cricket. It was harder for his sister, a charming 
intelligent woman who had never married and 
who had spent much of her life looking after him. 
With a humorous skill not unlike his own old 
form, she collected every scrap of cricket news she 
could find, though she had never learned anything 
about the game. 

Once or twice the sarcastic love of the human 
comedy came bursting out. Two or three weeks 
before his death, he heard from the Royal Society 
that he was to be given their highest honour, the 
Copley Medal. He gave his Mephistophelian grin, 
the first time I had seen it in full splendour in all 
those months. ' Now I know that I must be pretty 
near the end. When people hurry up to give you 
honorific things there is exactly one conclusion to 
be drawn. ' 

After I heard that, I think I visited him twice. 
The last time was four or five days before he died. 


There was an Indian test team playing in Australia, 
and we talked about them. 

It was in that same week that he told his sister : 
1 If I knew that I was going to die today, I think I 
should still want to hear the cricket scores. ' 

He managed something very similar. Each 
evening that week before she left him, she read a 
chapter from a history of Cambridge university 
cricket. One such chapter contained the last words 
he heard, for he died suddenly, in the early 



I am indebted for many valuable criticisms to 
Professor C. D. Broad and Dr G. P. Snow, each 
of whom read my original manuscript. I have 
incorporated the substance of nearly all of 
their suggestions in my text, and have so re- 
moved a good many crudities and obscurities. 
In one case I have dealt with them dif- 
ferently. My § 28 is based on a short article 
which I contributed to Eureka (the journal of 
the Cambridge Archimedean Society) early in 
the year, and I found it impossible to remodel 
what I had written so recently and with so 
much care. Also, if I had tried to meet such 
important criticisms seriously, I should have 
had to expand this section so much as to 
destroy the whole balance of my essay. I have 
therefore left it unaltered, but have added a 
short statement of the chief points made by 
my critics in a note at the end. 

G. H. H. 
18 July 1940 

It is a melancholy experience for a pro- 
fessional mathematician to find himself writing 
about mathematics. The function of a mathe- 
matician is to do something, to prove new 
theorems, to add to mathematics, and not to 
talk about what he or other mathematicians 
have done. Statesmen despise publicists, 
painters despise art-critics, and physiologists, 
physicists, or mathematicians have usually 
similar feelings; there is no scorn more pro- 
found, or on the whole more justifiable, than 
that of the men who make for the men who 
explain. Exposition, criticism, appreciation, 
is work for second-rate minds. 

I can remember arguing this point once in 
one of the few serious conversations that I ever 
had with Housman. Housman, in his Leslie 
Stephen lecture The Name and Nature of Poetry, 
had denied very emphatically that he was a 
'critic'; but he had denied it in what seemed 
to me a singularly perverse way, and had 

expressed an admiration for literary criticism 
which startled and scandalized me. 

He had begun with a quotation from his 
inaugural lecture, delivered twenty-two years 
before — 

Whether the faculty of literary criticism is the 
best gift that Heaven has in its treasuries, I cannot 
say ; but Heaven seems to think so, for assuredly it 
is the gift most charily bestowed. Orators and 
poets . . . , if rare in comparison with blackberries, 
are commoner than returns of Halley's comet: 
literary critics are less common. . . . 

And he had continued — 

In these twenty-two years I have improved in 
some respects and deteriorated in others, but I have 
not so much improved as to become a literary critic, 
nor so much deteriorated as to fancy that I have 
become one. 

It had seemed to me deplorable that a great 
scholar and a fine poet should write like this, 
and, finding myself next to him in Hall a few 
weeks later, I plunged in and said so. Did he 
really mean what he had said to be taken very 
seriously? Would the life of the best of critics 
really have seemed to him comparable with 
that of a scholar and a poet? We argued these 

questions all through dinner, and I think that 
finally he agreed with me. I must not seem to 
claim a dialectical triumph over a man who 
can no longer contradict me ; but ' Perhaps not 
entirely' was, in the end, his reply to the first 
question, and 'Probably no' to the second. 

There may have been some doubt about 
Housman's feelings, and I do not wish to claim 
him as on my side ; but there is no doubt at all 
about the feelings of men of science, and I 
share them fully. If then I find myself writing, 
not mathematics but 'about' mathematics, it 
is a confession of weakness, for which I may 
rightly be scorned or pitied by younger and 
more vigorous mathematicians. I write about 
mathematics because, like any other mathe- 
matician who has passed sixty, I have no 
longer the freshness of mind, the energy, or the 
patience to carry on effectively with my proper 

I propose to put forward an apology for 
mathematics ; and I may be told that it needs 
none, since there are now few studies more 


generally recognized, for good reasons or bad, 
as profitable and praiseworthy. This may be 
true; indeed it is probable, since the sensa- 
tional triumphs of Einstein, that stellar astro- 
nomy and atomic physics are the only sciences 
which stand higher in popular estimation. 
A mathematician need not now consider him- 
self on the defensive. He does not have to meet 
the sort of opposition described by Bradley in 
the admirable defence of metaphysics which 
forms the introduction to Appearance and Reality. 
A metaphysician, says Bradley, will be told 
that 'metaphysical knowledge is wholly im- 
possible', or that 'even if possible to a certain 
degree, it is practically no knowledge worth 
the name ' . ' The same problems,' he will hear, 
'the same disputes, the same sheer failure. 
Why not abandon it and come out? Is there 
nothing else more worth your labour?' There 
is no one so stupid as to use this sort of language 
about mathematics. The mass of mathematical 
truth is obvious and imposing; its practical 
applications, the bridges and steam-engines 
and dynamos, obtrude themselves on the 
dullest imagination. The public does not need 


to be convinced that there is something in 

All this is in its way very comforting to 
mathematicians, but it is hardly possible for 
a genuine mathematician to be content with it. 
Any genuine mathematician must feel that it 
is not on these crude achievements that the 
real case for mathematics rests, that the 
popular reputation of mathematics is based 
largely on ignorance and confusion, and that 
there is room for a more rational defence. At 
any rate, I am disposed to try to make one. 
It should be a simpler task, at any rate, than 
Bradley's difficult apology. 

I shall ask, then, why is it really worth while 
to make a serious study of mathematics ? What 
is the proper justification of a mathematician's 
life? And my answers will be, for the most 
part, such as are to be expected from a mathe- 
matician: I think that it is worth while, that 
there is ample justification. But I should say 
at once that, in defending mathematics, I shall 
be defending myself, and that my apology is 
bound to be to some extent egotistical. I should 
not think it worth while to apologize for my 


subject if I regarded myself as one of its 

Some egotism of this sort is inevitable, and 
I do not feel that it really needs justification. 
Good work is not done by 'humble' men. It 
is one of the first duties of a professor, for 
example, in any subject, to exaggerate a little 
both the importance of his subject and his own 
importance in it. A man who is always asking 
'Is what I do worth while?' and 'Am I the 
right person to do it? ' will always be ineffective 
himself and a discouragement to others. He 
must shut his eyes a little and think a little 
more of his subject and himself than they 
deserve. This is not too difficult: it is harder 
not to make his subject and himself ridiculous 
by shutting his eyes too tightly. 

A man who sets out to justify his existence and 
his activities has to distinguish two different 
questions. The first is whether the work which 
he does is worth doing; and the second is why 
he does it (whatever its value may be). 


The first question is often very difficult, and 
the answer very discouraging, but most people 
will find the second easy enough even then. 
Their answers, if they are honest, will usually 
take one or other of two forms; and the second 
form is merely a humbler variation of the first, 
which is the only answer which we need con- 
sider seriously. 

( i ) ' I do what I do because it is the one and 
only thing that I can do at all well. I am a 
lawyer, or a stockbroker, or a professional 
cricketer, because I have some real talent for 
that particular job. I am a lawyer because 
I have a fluent tongue, and am interested in 
legal subtleties; I am a stockbroker because 
my judgement of the markets is quick and 
sound; I am a professional cricketer because. 
I can bat unusually well. I agree that it might 
be better to be a poet or a mathematician, but 
unfortunately I have no talent for such pur- 

I am not suggesting that this is a defence 
which can be made by most people, since most 
people can do nothing at all well. But it is 
impregnable when it can be made without 


absurdity, as it can by a substantial minority : 
perhaps five or even ten per cent of men can 
do something rather well. It is a tiny minority 
who can do anything really well, and the 
number of men who can do two things well is 
negligible. If a man has any genuine talent, 
he should be ready to make almost any sacrifice 
in order to cultivate it to the full. 

This view was endorsed by Dr Johnson — 

When I told him that I had been to see [his 
namesake] Johnson ride upon three horses, he said 
'Such a man, sir, should be encouraged, for his 
performances show the extent of the human 
powers . . . ' — 

and similarly he would have applauded moun- 
tain climbers, channel swimmers, and blind- 
fold chess-players. For my own part, I am 
entirely in sympathy with all such attempts at 
remarkable achievement. I feel some sym- 
pathy even with conjurors and ventriloquists; 
and when Alekhine and Bradman set out to 
beat records, I am quite bitterly disappointed 
if they fail. And here both Dr Johnson and I 
find ourselves in agreement with the public. 
As W. J. Turner has said so truly, it is only the 


' highbrows ' (in the unpleasant sense) who do 
not admire the 'real swells'. 

We have of course to take account of the 
differences in value between different activi- 
ties. I would rather be a novelist or a painter 
than a statesman of similar rank; and there 
are many roads to fame which most of us 
would reject as actively pernicious. Yet it is 
seldom that such differences of value will turn 
the scale in a man's choice of a career, 
which will almost always be dictated by the 
limitations of his natural abilities. Poetry is 
more valuable than cricket, but Bradman 
would be a fool if he sacrificed his cricket 
in order to write second-rate minor poetry 
(and I suppose that it is unlikely that he 
could do better). If the cricket were a little 
less supreme, and the poetry better, then the 
choice might be more difficult : I do not know 
whether I would rather have been Victor 
Trumper or Rupert Brooke. It is fortunate 
that such dilemmas occur so seldom. 

I may add that they are particularly un- 
likely to present themselves to a mathema- 
tician. It is usual to exaggerate rather grossly 


the differences between the mental processes 
of mathematicians and other people, but it is 
undeniable that a gift for mathematics is one 
of the most specialized talents, and that 
mathematicians as a class are not particularly 
distinguished for general ability or versatility. 
If a man is in any sense a real mathematician, 
then it is a hundred to one that his mathematics 
will be far better than anything else he can do, 
and that he would be silly if he surrendered 
any decent opportunity of exercising his one 
talent in order to do undistinguished work 
in other fields. Such a sacrifice could be 
justified only by economic necessity or age. 

I had better say something here about this 
question of age, since it is particularly im- 
portant for mathematicians. No mathema- 
tician should ever allow himself to forget that 
mathematics, more than any other art or 
science, is a young man's game. To take a 
simple illustration at a comparatively humble 


level, the average age of election to the Royal 
Society is lowest in mathematics. 

We can naturally find much more striking 
illustrations. We may consider, for example, 
the career of a man who was certainly one of 
the world's three greatest mathematicians. 
Newton gave up mathematics at fifty, and had 
lost his enthusiasm long before ; he had recog- 
nized no doubt by the time that he was forty 
that his great creative days were over. His 
greatest ideas of all, fluxions and the law of 
gravitation, came to him about 1666, when he 
was twenty-four — ' in those days I was in the 
prime of my age for invention, and minded 
mathematics and philosophy more than at any 
time since'. He made big discoveries until he 
was nearly forty (the 'elliptic orbit' at thirty- 
seven), but after that he did little but polish 
and perfect. 

Galois died at twenty-one, Abel at twenty- 
seven, Ramanujan at thirty-three, Riemann 
at forty. There have been men who have done 
great work a good deal later; Gauss's great 
memoir on differential geometry was pub- 
lished when he was fifty (though he had had 


the fundamental ideas ten years before) . I do 
not know an instance of a major mathematical 
advance initiated by a man past fifty. If a man 
of mature age loses interest in and abandons 
mathematics, the loss is not likely to be very 
serious either for mathematics or for himself. 
On the other hand the gain is no more likely 
to be substantial ; the later records of mathe- 
maticians who have left mathematics are not 
particularly encouraging. Newton made a 
quite competent Master of the Mint (when he 
was not quarrelling with anybody) . Painleve 
was a not very successful Premier of France. 
Laplace's political career was highly dis- 
creditable, but he is hardly a fair instance, 
since he was dishonest rather than incom- 
petent, and never really 'gave up' mathe- 
matics. There is no instance, so far as I know, 
of a first-rate mathematician abandoning 
mathematics and attaining first-rate distinc- 
tion in any other field. There may have 
been young men who would have been first- 
rate mathematicians if they had stuck to 
mathematics, but I have never heard of a 
really plausible example. And all this is fully 


borne out by my own very limited experience. 
Every young mathematician of real talent 
whom I have known has been faithful to 
mathematics, and not from lack of ambition 
but from abundance of it; they have all 
recognized that there, if anywhere, lay the 
road to a life of any distinction. 

There is also what I called the 'humbler 
variation' of the standard apology; but I may 
dismiss this in a very few words. 

(2) 'There is nothing that I can do par- 
ticularly well. I do what I do because it came 
my way. I really never had a chance of doing 
anything else.' And this apology too I accept 
as conclusive. It is quite true that most people 
can do nothing well. If so, it matters very 
little what career they choose, and there is 
really nothing more to say about it. It is a con- 
clusive reply, but hardly one likely to be made 
by a man with any pride; and I may assume 
that none of us would be content with it. 


It is time to begin thinking about the first 
question which I put in § 3, and which is so 
much more difficult than the second. Is 
mathematics, what I and other mathema- 
ticians mean by mathematics, worth doing; 
and if so, why? 

I have been looking again at the first pages 
of the inaugural lecture which I gave at 
Oxford in 1920, where there is an outline of 
an apology for mathematics. It is very in- 
adequate (less than a couple of pages), and it 
is written in a style (a first essay, I suppose, in 
what I then imagined to be the 'Oxford 
manner') of which I am not now particularly 
proud; but I still feel that, however much 
development it may need, it contains the 
essentials of the matter. I will resume what I 
said then, as a preface to a fuller discussion. 

( 1 ) I began by laying stress on the harmless- 
ness of mathematics — 'the study of mathe- 
matics is, if an unprofitable, a perfectly 
harmless and innocent occupation'. I shall 


stick to that, but obviously it will need a good 
deal of expansion and explanation. 

Is mathematics 'unprofitable'? In some 
ways, plainly, it is not; for example, it gives 
great pleasure to quite a large number of 
people. I was using the word, however, in 
a narrower sense — is mathematics 'useful', 
directly useful, as other sciences such as chem- 
istry and physiology are? This is not an 
altogether easy or uncontroversial question, 
and I shall ultimately say No, though some 
mathematicians, and most outsiders, would no 
doubt say Yes. And is mathematics 'harm- 
less'? Again the answer is not obvious, and 
the question is one which I should have in 
some ways preferred to avoid, since it raises 
the whole problem of the effect of science on 
war. Is mathematics harmless, in the sense in 
which, for example, chemistry plainly is not? 
I shall have to come back to both these 
questions later. 

(2) I went on to say that 'the scale of the 
universe is large and, if we are wasting our 
time, the waste of the lives of a few university 
dons is no such overwhelming catastrophe': 


and here I may seem to be adopting, or 
affecting, the pose of exaggerated humility 
which I repudiated a moment ago. I am sure 
that that was not what was really in my mind ; 
I was trying to say in a sentence what I have 
said at much greater length in § 3. I was 
assuming that we dons really had our little 
talents, and that we could hardly be wrong if 
we did our best to cultivate them fully. 

(3) Finally (in what seem to me now some 
rather painfully rhetorical sentences) I em- 
phasized the permanence of mathematical 
achievement — 

What we do may be small, but it has a certain 
character of permanence; and to have produced 
anything of the slightest permanent interest, whether 
it be a copy of verses or a geometrical theorem, is 
to have done something utterly beyond the powers 
of the vast majority of men. 


In these days of conflict between ancient and 
modern studies, there must surely be something 
to be said for a study which did not begin with 
Pythagoras, and will not end with Einstein, but is 
the oldest and the youngest of all. 


All this is ' rhetoric ' ; but the substance of it 
seems to me still to ring true, and I can expand 
it at once without prejudging any of the other 
questions which I am leaving open. 

I shall assume that I am writing for readers 
who are full, or have in the past been full, of 
a proper spirit of ambition. A man's first duty, 
a young man's at any rate, is to be ambitious. 
Ambition is a noble passion which may 
legitimately take many forms; there was 
something noble in the ambition of Attila 
or Napoleon: but the noblest ambition is 
that of leaving behind one something of 
permanent value — 

Here on the level sand, 
Between the sea and land, 
What shall I build or write 
Against the fall of night? 

Tell me of runes to grave 
That hold the bursting wave, 
Or bastions to design 
For longer date than mine. 


Ambition has been the driving force behind 
nearly all the best work of the world. In 
particular, practically all substantial contri- 
butions to human happiness have been made 
by ambitious men. To take two famous 
examples, were not Lister and Pasteur am- 
bitious? Or, on a humbler level, King Gillette 
and William Willett ; and who in recent times 
have contributed more to human comfort 
than they? 

Physiology provides particularly good ex- 
amples, just because it is so obviously a 
'beneficial' study. We must guard against a 
fallacy common among apologists of science, 
a fallacy into which, for example, Professor 
A. V. Hill has fallen, the fallacy of supposing 
that the men whose work most benefits hu- 
manity are thinking much of that while they 
do it, that physiologists, in short, have par- 
ticularly noble souls. A physiologist may 
indeed be glad to remember that his work 
will benefit mankind, but the motives which 
provide the force and the inspiration for it are 
indistinguishable from those of a classical 
scholar or a mathematician. 


There are many highly respectable motives 
which may lead men to prosecute research, 
but three which are much more important 
than the rest. The first (without which the rest 
must come to nothing) is intellectual curiosity, 
desire to know the truth. Then, professional 
pride, anxiety to be satisfied with one's per- 
formance, the shame that overcomes any 
self-respecting craftsman when his work is 
unworthy of his talent. Finally, ambition, 
desire for reputation, and the position, even 
the power or the money, which it brings. It 
may be fine to feel, when you have done your 
work, that you have added to the happiness 
or alleviated the sufferings of others, but that 
will not be why you did it. So if a mathema- 
tician, or a chemist, or even a physiologist, 
were to tell me that the driving force in his 
work had been the desire to benefit humanity, 
then I should not believe him (nor should I 
think the better of him if I did) . His dominant 
motives have been those which I have stated, 
and in which, surely, there is nothing of which 
any decent man need be ashamed. 



If intellectual curiosity, professional pride, 
and ambition are the dominant incentives to 
research, then assuredly no one has a fairer 
chance of gratifying them than a mathema- 
tician. His subject is the most curious of all — 
there is none in which truth plays such odd 
pranks. It has the most elaborate and the 
most fascinating technique, and gives un- 
rivalled openings for the display of sheer 
professional skill. Finally, as history proves 
abundantiy, mathematical achievement, what- 
ever its intrinsic worth, is the most enduring 
of all. 

We can see this even in semi-historic 
civilizations. The Babylonian and Assyrian 
civilizations have perished ; Hammurabi, Sar- 
gon, and Nebuchadnezzar are empty names; 
yet Babylonian mathematics is still interesting, 
and the Babylonian scale of 60 is still used in 
astronomy. But of course the crucial case is 
that of the Greeks. 

The Greeks were the first mathematicians 

who are still 'real' to us to-day. Oriental 
mathematics may be an interesting curiosity, 
but Greek mathematics is the real thing. The 
Greeks first spoke a language which modern 
mathematicians can understand; as Little- 
wood said to me once, they are not clever 
schoolboys or 'scholarship candidates', but 
' Fellows of another college '. So Greek mathe- 
matics is 'permanent', more permanent even 
than Greek literature. Archimedes will be 
remembered when Aeschylus is forgotten, 
because languages die and mathematical ideas 
do not. 'Immortality' may be a silly word, 
but probably a mathematician has the best 
chance of whatever it may mean. 

Nor need he fear very seriously that the 
future will be unjust to him. Immortality is 
often ridiculous or cruel: few of us would have 
chosen to be Og or Ananias or Gallio. Even 
in mathematics, history sometimes plays 
strange tricks ; Rolle figures in the text-books 
of elementary calculus as if he had been a 
mathematician like Newton; Farey is immortal 
because he- failed to understand a theorem 
which Haros had proved perfectly fourteen 


years before; the names of five worthy Nor- 
wegians still stand in Abel's Life, just for one 
act of conscientious imbecility, dutifully per- 
formed at the expense of their country's 
greatest man. But on the whole the history of 
science is fair, and this is particularly true in 
mathematics. No other subject has such clear- 
cut or unanimously accepted standards, and 
the men who are remembered are almost 
always the men who merit it. Mathematical 
fame, if you have the cash to pay for it, is one 
of the soundest and steadiest of investments. 


All this is very comforting for dons, and 
especially for professors of mathematics. It is 
sometimes suggested, by lawyers or politicians 
or business men, that an academic career is 
one sought mainly by cautious and unambi- 
tious persons who care primarily for comfort 
and security. The reproach is quite misplaced. 
A don surrenders something, and in par- 
ticular the chance of making large sums of 
money — it is very hard for a professor to make 

£2000 a year; and security of tenure is 
naturally one of the considerations which 
make this particular surrender easy. That is 
not why Housman would have refused to be 
Lord Simon or Lord Beaverbrook. He would 
have rejected their careers because of his 
ambition, because he would have scorned to 
be a man to be forgotten in twenty years. 

Yet how painful it is to feel that, with all 
these advantages, one may fail. I can remem- 
ber Bertrand Russell telling me of a horrible 
dream. He was in the top floor of the Uni- 
versity Library, about a.d. 2100. A library 
assistant was going round the shelves carrying 
an enormous bucket, taking down book after 
book, glancing at them, restoring them to the 
shelves or dumping them into the bucket. At 
last he came to three large volumes which 
Russell could recognize as the last surviving 
copy of Principia mathematica. He took down 
one of the volumes, turned over a few pages, 
seemed puzzled for a moment by the curious 
symbolism, closed the volume, balanced it in 
his hand and hesitated 



A mathematician, like a painter or a poet, 
is a maker of patterns. If his patterns are more 
permanent than theirs, it is because they are 
made with ideas. A painter makes patterns 
with shapes and colours, a poet with words. 
A painting may embody an 'idea', but the 
idea is usually commonplace and unimport- 
ant. In poetry, ideas count for a good deal 
more; but, as Housman insisted, the im- 
portance of ideas in poetry is habitually 
exaggerated: 'I cannot satisfy myself that 

there are any such things as poetical ideas 

Poetry is not the thing said but a way of saying 

Not all the water in the rough rude sea 
Can wash the balm from an anointed King. 

Could lines be better, and could ideas be at 
once more trite and more false? The poverty 
of the ideas seems hardly to affect the beauty 
of the verbal pattern. A mathematician, on 
the other hand, has no material to work with 


but ideas, and so his patterns are likely to last 
longer, since ideas wear less with time than 

The mathematician's patterns, like the 
painter's or the poet's, must be beautiful; the 
ideas, like the colours or the words, must fit 
together in a harmonious way. Beauty is the 
first test: there is no permanent place in the 
world for ugly mathematics. And here I must 
deal with a misconception which is still wide- 
spread (though probably much less so now 
than it was twenty years ago) , what Whitehead 
has called the ' literary superstition ' that love 
of and aesthetic appreciation of mathematics 
is ' a monomania confined to a few eccentrics 
in each generation'. 

It would be difficult now to find an educated 
man quite insensitive to the aesthetic appeal 
of mathematics. It may be very hard to define 
mathematical beauty, but that is just as true 
of beauty of any kind — we may not know quite 
what we mean by a beautiful poem, but that 
does not prevent us from recognizing one when 
we read it. Even Professor Hogben, who is 
out to minimize at all costs the importance of 


the aesthetic element in mathematics, does not 
venture to deny its reality. ' There are, to be 
sure, individuals for whom mathematics exer- 
cises a coldly impersonal attraction The 

aesthetic appeal of mathematics may be very 
real for a chosen few.' But they are 'few', he 
suggests, and they feel 'coldly' (and are really 
rather ridiculous people, who live in silly little 
university towns sheltered from the fresh 
breezes of the wide open spaces). In this he is 
merely echoing Whitehead's 'literary super- 

The fact is that there are few more ' popular ' 
subjects than mathematics. Most people have 
some appreciation of mathematics, just as most 
people can enjoy a pleasant tune; and there 
are probably more people really interested in 
mathematics than in music. Appearances may 
suggest the contrary, but there are easy ex- 
planations. Music can be used to stimulate 
mass emotion, while mathematics cannot; and 
musical incapacity is recognized (no doubt 
rightly) as mildly discreditable, whereas most 
people are so frightened of the name of 
mathematics that they are ready, quite un- 

affectedly, to exaggerate their own mathe- 
matical stupidity. 

A very little reflection is enough to expose 
the absurdity of the 'literary superstition'. 
There are masses of chess-players in every 
civilized country — in Russia, almost the whole 
educated population; and every chess-player 
can recognize and appreciate a 'beautiful' 
game or problem. Yet a chess problem is simply 
an exercise in pure mathematics (a game not 
entirely, since psychology also plays a part), 
and everyone who calls a problem ' beautiful ' 
is applauding mathematical beauty, even if 
it is beauty of a comparatively lowly kind. 
Chess problems are the hymn- tunes of mathe- 

We may learn the same lesson, at a lower 
level but for a wider public, from bridge, or 
descending further, from the puzzle columns 
of the popular newspapers. Nearly all their 
immense popularity is a tribute to the drawing 
power of rudimentary mathematics, and the 
better makers of puzzles, such as Dudeney or 
' Caliban ', use very little else. They know their 
business; what the public wants is a little 


intellectual 'kick', and nothing else has quite 
the kick of mathematics. 

I might add that there is nothing in the 
world which pleases even famous men (and 
men who have used disparaging language 
about mathematics) quite so much as to dis- 
cover, or rediscover, a genuine mathematical 
theorem. Herbert Spencer republished in his 
autobiography a theorem about circles which 
he proved when he was twenty (not knowing 
that it had been proved over two thousand 
years before by Plato). Professor Soddy is a 
more recent and a more striking example (but 
his theorem really is his own)*. 


A chess problem is genuine mathematics, but 
it is in some way ' trivial ' mathematics. How- 
ever ingenious and intricate, however original 
and surprising the moves, there is something 
essential lacking. Chess problems are unim- 

* See his letters on the 'Hcxlet' in Nature, vols. 137-9 


portant. The best mathematics is serious as well 
as beautiful — 'important' if you like, but the 
word is very ambiguous, and 'serious' expresses 
what I mean much better. 

I am not thinking of the ' practical ' con- 
sequences of mathematics. I have to return to 
that point later: at present I will say only that 
if a chess problem is, in the crude sense, 
'useless', then that is equally true of most of 
the best mathematics ; that very little of mathe- 
matics is useful practically, and that that little 
is comparatively dull. The 'seriousness' of a 
mathematical theorem lies, not in its practical 
consequences, which are usually negligible, 
but in the significance of the mathematical ideas 
which it connects. We may say, roughly, that 
a mathematical idea is 'significant* if it can be 
connected, in a natural and illuminating way, 
with a large complex of other mathematical 
ideas. Thus a serious mathematical theorem, 
a theorem which connects significant ideas, is 
likely to lead to important advances in mathe- 
matics itself and even in other sciences. No 
chess problem has ever affected the general 
development of scientific thought; Pythagoras, 


Newton, Einstein have in their times changed 
its whole direction. 

The seriousness of a theorem, of course, does 
not lie in its consequences, which are merely 
the evidence for its seriousness. Shakespeare 
had an enormous influence on the develop- 
ment of the English language, Otway next to 
none, but that is not why Shakespeare was the 
better poet. He was the better poet because 
he wrote much better poetry. The inferiority 
of the chess problem, like that of Otway's 
poetry, lies not in its consequences but in its 

There is one more point which I shall dis- 
miss very shortly, not because it is uninteresting 
but because it is difficult, and because I have 
no qualifications for any serious discussion in 
aesthetics. The beauty of a mathematical 
theorem depends a great deal on its seriousness, 
as even in poetry the beauty of a line may 
depend to some extent on the significance of 
the ideas which it contains. I quoted two lines 
of Shakespeare as an example of the sheer 
beauty of a verbal pattern ; but 

After life's fitful fever he sleeps well 


seems still more beautiful. The pattern is just 
as fine, and in this case the ideas have signi- 
ficance and the thesis is sound, so that our 
emotions are stirred much more deeply. The 
ideas do matter to the pattern, even in poetry, 
and much more, naturally, in mathematics; but 
I must not try to argue the question seriously. 


It will be clear by now that, if we are to have 
any chance of making progress, I must produce 
examples of 'real' mathematical theorems, 
theorems which every mathematician will 
admit to be first-rate. And here I am very 
heavily handicapped by the restrictions under 
which I am writing. On the one hand my 
examples must be very simple, and intelligible 
to a reader who has no specialized mathe- 
matical knowledge ; no elaborate preliminary 
explanations must be needed; and a reader 
must be able to follow the proofs as well as the 
enunciations. These conditions exclude, for 
instance, many of the most beautiful theorems 
of the theory of numbers, such as Fermat's ' two 


square ' theorem or the law of quadratic re- 
ciprocity. And on the other hand my examples 
should be drawn from 'pukka' mathematics, 
the mathematics of the working professional 
mathematician; and this condition excludes 
a good deal which it would be comparatively 
easy to make intelligible but which trespasses 
on logic and mathematical philosophy. 

I can hardly do better than go back to the 
Greeks. I will state and prove two of the 
famous theorems of Greek mathematics. They 
are ' simple ' theorems, simple both in idea and 
in execution, but there is no doubt at all about 
their being theorems of the highest class. Each 
is as fresh and significant as when it was dis- 
covered — two thousand years have not written 
a wrinkle on either of them. Finally, both the 
statements and the proofs can be mastered in 
an hour by any intelligent reader, however 
slender his mathematical equipment. 

i. The first is Euclid's* proof of the exist- 
ence of an infinity of prime numbers. 

* Elements rx 20. The real origin of many theorems in 
the Elements is obscure, but there seems to be no particular 
reason for supposing that this one is not Euclid's own. 


The prime numbers or primes are the numbers 
(A) 2,3,5,7, 11, 13, 17, 19,23,29, ... 
which cannot be resolved into smaller factors* . 
Thus 37 and 317 are prime. The primes are 
the material out of which all numbers are built 
up by multiplication: thus 666 = 
Every number which is not prime itself is 
divisible by at least one prime (usually, of 
course, by several). We have to prove that 
there are infinitely many primes, i.e. that the 
series (A) never comes to an end. 

Let us suppose that it does, and that 

2, 3>5, ••• >P 
is the complete series (so that P is the largest 
prime); and let us, on this hypothesis, con- 
sider the number 

£=(2.3.5 f) + l. 

It is plain that Q_ is not divisible by any of 
2, 3, 5, •••> P\ fo r ^ leaves the remainder 1 
when divided by any one of these numbers. 
But, if not itself prime, it is divisible by some 
prime, and therefore there is a prime (which 

* There are technical reasons for not counting 1 as a 


may be Q, itself) greater than any of them. 
This contradicts our hypothesis, that there is 
no prime greater than P; and therefore this 
hypothesis is false. 

The proof is by reductio ad absurdum, and 
reductio ad absurdum, which Euclid loved so 
much, is one of a mathematician's finest 
weapons*. It is a far finer gambit than any 
chess gambit: a chess player may offer the 
sacrifice of a pawn or even a piece, but a 
mathematician offers the game. 

l 3 

2. My second example is Pythagoras'sf 
proof of the 'irrationality' of J2. 

A 'rational number' is a fraction r , where 

a and b are integers; we may suppose that a 

and b have no common factor, since if they 

had we could remove it. To say that '^2 is 

irrational ' is merely another way of saying 

* The proof can be arranged so as to avoid a reductio, and 
logicians of some schools would prefer that it should be. 

f The proof traditionally ascribed to Pythagoras, and 
certainly a product of his school. The theorem occurs, in a 
much more general form, in Euclid {Elements x 9). 


that 2 cannot be expressed in the form \j) ', 

and this is the same thing as saying that the 

(B) a 2 = 2b 2 

cannot be satisfied by integral values of a and b 
which have no common factor. This is a 
theorem of pure arithmetic, which does not 
demand any knowledge of 'irrational numbers' 
or depend on any theory about their nature. 
We argue again by reductio ad absurdum ; we 
suppose that (B) is true, a and b being integers 
without any common factor. It follows from 

(B) that a 2 is even (since 2b 2 is divisible by 2), 
and therefore that a is even (since the square 
of an odd number is odd;. If a is even then 

(C) a = 2c 

for some integral value oft; and therefore 

2b 2 = a 2 = (2c) 2 = 4.C 2 

(D) b 2 = 2c 2 . 

Hence b 2 is even, and therefore (for the same 
reason as before) b is even. That is to say, 


a and b are both even, and so have the common 
factor 2. This contradicts our hypothesis, and 
therefore the hypothesis is false. 

It follows from Pythagoras's theorem that 
the diagonal of a square is incommensurable 
with the side (that their ratio is not a rational 
number, that there is no unit of which both are 
integral multiples). For if we take the side as 
our unit of length, and the length of the 
diagonal is d, then, by a very familiar theorem 
also ascribed to Pythagoras*, 
d 2 = I 2 +I 2 =;2j 

so that d cannot be a rational number. 

I could quote any number of fine theorems 
from the theory of numbers whose meaning 
anyone can understand. For example, there 
is what is called ' the fundamental theorem of 
arithmetic', that any integer can be resolved, 
in one way only, into a product of primes. Thus 
666 =, and there is no other decom- 
position ; it is impossible that 666 = 2 . 1 1 . 29 
or that 13.89= 17.73 (and we can see so 
without working out the products). This 

* Euclid, Elements i 47. 

theorem is, as its name implies, the foundation 
of higher arithmetic ; but the proof, although 
not 'difficult', requires a certain amount of 
preface and might be found tedious by an 
unmathematical reader. 

Another famous and beautiful theorem is 
Fermat's 'two square' theorem. The primes 
may (if we ignore the special prime 2) be 
arranged in two classes; the primes 

5, 13, 17.29,37,41, ... 

which leave remainder 1 when divided by 4, 
and the primes 

3>7> "s 19,23,31, ... 
which leave remainder 3. All the primes of 
the first class, and none of the second, can be 
expressed as the sum of two integral squares : 

— t2 

+ 2 2 , i3 = 2 2 + 3 2 , 

i7=i 2 + 4 2 , 2 9 = 2 2 + 5 2 ; 

but 3, 7, 11, and 19 are not expressible in 
this way (as the reader may check by trial). 
This is Fermat's theorem, which is ranked, 
very justly, as one of the finest of arithmetic. 
Unfortunately there is no proof within the 


comprehension of anybody but a fairly expert 

There are also beautiful theorems in the 
'theory of aggregates ' (Mengenlehre), such as 
Cantor's theorem of the ' non-enumerability ' 
of the continuum. Here there is just the 
opposite difficulty. The proof is easy enough, 
when once the language has been mastered, 
but considerable explanation is necessary 
before the meaning of the theorem becomes 
clear. So I will not try to give more examples. 
Those which I have given are test cases, and 
a reader who cannot appreciate them is un- 
likely to appreciate anything in mathematics. 

I said that a mathematician was a maker of 
patterns of ideas, and that beauty and serious- 
ness were the criteria by which his patterns 
should be judged. I can hardly believe that 
anyone who has understood the two theorems 
will dispute that they pass these tests. If we 
compare them with Dudeney's most ingenious 
puzzles, or the finest chess problems that 
masters of that art have composed, their 
superiority in both respects stands out: there 
is an unmistakable difference of class. They 


are much more serious, and also much more 
beautiful ; can we define, a little more closely, 
where their superiority lies? 


In the first place, the superiority of the 
mathematical theorems in seriousness is obvious 
and overwhelming. The chess problem is the 
product of an ingenious but very limited 
complex of ideas, which do not differ from one 
another very fundamentally and have no 
external repercussions. We should think in the 
same way if chess had never been invented, 
whereas the theorems of Euclid and Pythagoras 
have influenced thought profoundly, even 
outside mathematics. 

Thus Euclid's theorem is vital for the whole 
structure of arithmetic. The primes are the 
raw material out of which we have to build 
arithmetic, and Euclid's theorem assures us 
that we have plenty of material for the task. 
But the theorem of Pythagoras has wider 
applications and provides a better text. 

We should observe first that Pythagoras's 


argument is capable of far-reaching extension, 
and can be applied, with little change of 
principle, to very wide classes of 'irrationals'. 
We can prove very similarly (as Theaetetus 
seems to have done) that 

^3> J$ 9 ^7> Jii, >/i3> ^7 
are irrational, or (going beyond Theaetetus) 
that ^2 and Uvj are irrational*. 

Euclid's theorem tells us that we have a good 
supply of material for the construction of a 
coherent arithmetic of the integers. Pytha- 
goras's theorem and its extensions tell us that, 
when we have constructed this arithmetic, it 
will not prove sufficient for our needs, since 
there will be many magnitudes which obtrude 
themselves upon our attention and which it 
will be unable to measure; the diagonal of the 
square is merely the most obvious example. 
The profound importance of this discovery 
was recognized at once by the Greek mathe- 
maticians. They had begun by assuming (in 

* See Ch. iv of Hardy and Wright's Introduction to the 
Theory of Numbers, where there are discussions of different 
generalizations of Pythagoras 's argument, and of a historical 
puzzle about Theaetetus. 


accordance, I suppose, with the 'natural' 
dictates of 'common sense') that all magni- 
tudes of the same kind are commensurable, 
that any two lengths, for example, are mul- 
tiples of some common unit, and they had 
constructed a theory of proportion based on 
this assumption. Pythagoras's discovery ex- 
posed the unsoundness of this foundation, and 
led to the construction of the much more 
profound theory of Eudoxus which is set out 
in the fifth book of the Elements, and which is 
regarded by many modern mathematicians as 
the finest achievement of Greek mathematics. 
This theory is astonishingly modern in spirit, 
and may be regarded as the beginning of the 
modern theory of irrational number, which 
has revolutionized mathematical analysis and 
had much influence on recent philosophy. 

There is no doubt at all, then, of the 
'seriousness' of either theorem. It is therefore 
the better worth remarking that neither 
theorem has the slightest 'practical' im- 
portance. In practical applications we are 
concerned only with comparatively small 
numbers; only stellar astronomy and atomic 


physics deal with 'large' numbers, and they 
have very little more practical importance, as 
yet, than the most abstract pure mathematics. 
I do not know what is the highest degree of 
accuracy which is ever useful to an engineer — 
we shall be very generous if we say ten signi- 
ficant figures. Then 


(the value of n to eight places of decimals) is 

the ratio c 



of two numbers of ten digits. The number of 
primes less than 1,000,000,000 is 50,847,478: 
that is enough for an engineer, and he can be 
perfectly happy without the rest. So much for 
Euclid's theorem; and, as regards Pythagoras's, 
it is obvious that irrationals are uninteresting 
to an engineer, since he is concerned only with 
approximations, and all approximations are 



A 'serious' theorem is a theorem which 
contains ' significant ' ideas, and I suppose that 
I ought to try to analyse a little more closely 
the qualities which make a mathematical idea 
significant. This is very difficult, and it is un- 
likely that any analysis which I can give will be 
very valuable. We can recognize a ' significant ' 
idea when we see it, as we can those which 
occur in my two standard theorems ; but this 
power of recognition requires a rather high 
degree of mathematical sophistication, and 
of that familiarity with mathematical ideas 
which comes only from many years spent in 
their company. So I must attempt some sort 
of analysis ; and it should be possible to make 
one which, however inadequate, is sound and 
intelligible so far as it goes. There are two 
things at any rate which seem essential, a 
certain generality and a certain depth; but 
neither quality is easy to define at all 


A significant mathematical idea, a serious 
mathematical theorem, should be 'general' 
in some such sense as this. The idea should 
be one which is a constituent in many mathe- 
matical constructs, which is used in the proof 
of theorems of many different kinds. The 
theorem should be one which, even if stated 
originally (like Pythagoras's theorem) in a 
quite special form, is capable of considerable 
extension and is typical of a whole class of 
theorems of its kind. The relations revealed by 
the proof should be such as connect many 
different mathematical ideas. All this is very 
vague, and subject to many reservations. But 
it is easy enough to see that a theorem is un- 
likely to be serious when it lacks these qualities 
conspicuously; we have only to take examples 
from the isolated curiosities in which arith- 
metic abounds. I take two, almost at random, 
from Rouse Ball's Mathematical Recreations*. 

(a) 8712 and 9801 are the only four-figure 
numbers which are integral multiples of their 
1 reversals ' : 

8712 = 4.2178, 9801=9.1089, 

* 1 ah edition, 1939 (revised by H. S. M. Coxeter). 

and there are no other numbers below 10,000 
which have this property. 

(b) There are just four numbers (after 1) 
which are the sums of the cubes of their digits, 

J 53 = i 3 -t-5 3 + 3 3 > 37o = 3 3 + 7 3 +o 3 > 
37 1 =3 s +7 3 + l3 > 4°7 = 4 3 +o 3 + 7 3 - 
These are odd facts, very suitable for puzzle 
columns and likely to amuse amateurs, but 
there is nothing in them which appeals much 
to a mathematician. The proofs are neither 
difficult nor interesting — merely a little tire- 
some. The theorems are not serious ; and it is 
plain that one reason (though perhaps not the 
most important) is the extreme speciality of 
both the enunciations and the proofs, which 
are not capable of any significant generaliza- 


'Generality' is an ambiguous and rather 
dangerous word, and we must be careful not 
to allow it to dominate our discussion too 
much. It is used in various senses both in 

mathematics and in writings about mathe- 
matics, and there is one of these in particular, 
on which logicians have very properly laid 
great stress, which is entirely irrelevant here. 
In this sense, which is quite easy to define, all 
mathematical theorems are equally and com- 
pletely 'general'. 

' The certainty of mathematics ', says White- 
head*, 'depends on its complete abstract 
generality.' When we assert that 2 + 3 = 5, 
we are asserting a relation between three 
groups of ' things ' ; and these ' things ' are not 
apples or pennies, or things of any one par- 
ticular sort or another, but just things, ' any 
old things'. The meaning of the statement is 
entirely independent of the individualities of 
the members of the groups. All mathematical 
'objects' or 'entities' or 'relations', such as 
' 2 ', ' 3 ', ' 5 ', ' + ', or ' = ', and all mathematical 
propositions in which they occur, are com- 
pletely general in the sense of being completely 
abstract. Indeed one of Whitehead's words 
is superfluous, since generality, in this sense, 
is abstractness. 

* Science and the Modern World, p. 33. 

This sense of the word is important, and the 
logicians are quite right to stress it, since it 
embodies a truism which a good many people 
who ought to know better are apt to forget. It is 
quite common, for example, for an astronomer 
or a physicist to claim that he has found a 
'mathematical proof that the physical uni- 
verse must behave in a particular way. All 
such claims, if interpreted literally, are strictly 
nonsense. It cannot be possible to prove 
mathematically that there will be an eclipse 
to-morrow, because eclipses, and other phy- 
sical phenomena, do not form part of the 
abstract world of mathematics; and this, I 
suppose, all astronomers would admit when 
pressed, however many eclipses they may have 
predicted correctly. 

It is obvious that we are not concerned with 
this sort of 'generality' now. We are looking 
for differences of generality between one mathe- 
matical theorem and another, and in White- 
head's sense all are equally general. Thus the 
'trivial' theorems (a) and (b) of § 15 are just 
as 'abstract' or 'general' as those of Euclid 
and Pythagoras, and so is a chess problem. It 


makes no difference to a chess problem whether 
the pieces are white and black, or red and 
green, or whether there are physical ' pieces ' 
at all ; it is the same problem which an expert 
carries easily in his head and which we have 
to reconstruct laboriously with the aid of the 
board. The board and the pieces are mere 
devices to stimulate our sluggish imaginations, 
and are no more essential to the problem than 
the blackboard and the chalk are to the 
theorems in a mathematical lecture. 

It is not this kind of generality, common to 
all mathematical theorems, which we are 
looking for now, but the more subtle and 
elusive kind of generality which I tried to 
describe in rough terms in § 15. And we must 
be careful not to lay too much stress even on 
generality of this kind (as I think logicians like 
Whitehead tend to do) . It is not mere * piling 
of subtlety of generalization upon subtlety of 
generalization'* which is the outstanding 
achievement of modern mathematics. Some 
measure of generality must be present in any 
high-class theorem, but too much tends in- 

* Science and the Modern World, p. 44. 

evitably to insipidity. ' Everything is what it 
is, and not another thing', and the differences 
between things are quite as interesting as their 
resemblances. We do not choose our friends 
because they embody all the pleasant qualities 
of humanity, but because they are the people 
that they are. And so in mathematics; a 
property common to too many objects can 
hardly be very exciting, and mathematical 
ideas also become dim unless they have plenty 
of individuality. Here at any rate I can quote 
Whitehead on my side: 'it is the large 
generalization, limited by a happy particu- 
larity, which is the fruitful conception*.' 


The second quality which I demanded in a 
significant idea was depth, and this is still more 
difficult to define. It has something to do with 
difficulty', the 'deeper' ideas are usually the 
harder to grasp : but it is not at all the same. 
The ideas underlying Pythagoras's theorem 
and its generalizations are quite deep, but no 

* Science and the Modern World, p. 46. 

mathematician now would find them difficult. 
On the other hand a theorem may be essen- 
tially superficial and yet quite difficult to 
prove (as are many ' Diophantine ' theorems, 
i.e. theorems about the solution of equations in 
integers) . 

It seems that mathematical ideas are ar- 
ranged somehow in strata, the ideas in each 
stratum being linked by a complex of relations 
both among themselves and with those above 
and below. The lower the stratum, the deeper 
(and in general the more difficult) the idea. 
Thus the idea of an 'irrational' is deeper than 
that of an integer ; and Py thagoras's theorem 
is, for that reason, deeper than Euclid's. 

Let us concentrate our attention on the 
relations between the integers, or some other 
group of objects lying in some particular 
stratum. Then it may happen that one of these 
relations can be comprehended completely, 
that we can recognize and prove, for example, 
some property of the integers, without any 
knowledge of the contents of lower strata. Thus 
we proved Euclid's theorem by consideration 
of properties of integers only. But there are 

1 10 

also many theorems about integers which we 
cannot appreciate properly, and still less 
prove, without digging deeper and considering 
what happens below. 

It is easy to find examples in the theory 
of prime numbers. Euclid's theorem is very 
important, but not very deep: we can prove 
that there are infinitely many primes without 
using any notion deeper than that of ' divisi- 
bility'. But new questions suggest themselves 
as soon as we know the answer to this one. 
There is an infinity of primes, but how is this 
infinity distributed ? Given a large number JV, 
say io 80 or io 1(}10 ,* about how many primes 
are there less than jV?f When we ask these 
questions, we find ourselves in a quite different 
position. We can answer them, with rather 
surprising accuracy, but only by boring much 
deeper, leaving the integers above us for a 
while, and using the most powerful weapons 
of the modern theory of functions. Thus the 

* It is supposed that the number of protons in the universe 

is about io 80 . The number io 10 , if written at length, would 
occupy about 50,000 volumes of average size. 

| As I mentioned in § 14, there are 50,847,478 primes less 
than 1,000,000,000; but that is as far as our exact knowledge 

I I I 

theorem which answers our questions (the 
so-called ' Prime Number Theorem') is a much 
deeper theorem than Euclid's or even Pytha- 

I could multiply examples, but this notion 
of 'depth' is an elusive one even for a mathe- 
matician who can recognize it, and I can 
hardly suppose that I could say anything more 
about it here which would be of much help to 
other readers. 


There is still one point remaining over from 
§ II, where I started the comparison between 
'real mathematics' and chess. We may take it 
for granted now that in substance, seriousness, 
significance, the advantage of the real mathe- 
matical theorem is overwhelming. It is almost 
equally obvious, to a trained intelligence, that 
it has a great advantage in beauty also; but 
this advantage is much harder to define or 
locate, since the main defect of the chess 
problem is plainly its 'triviality', and the 
contrast in this respect mingles with and dis- 

1 12 

turbs any more purely aesthetic judgement. 
What ' purely aesthetic ' qualities can we dis- 
tinguish in such theorems as Euclid's and 
Pythagoras's? I will not risk more than a few 
disjointed remarks. 

In both theorems (and in the theorems, of 
course, I include the proofs) there is a very 
high degree of unexpectedness, combined with 
inevitability and economy. The arguments take 
so odd and surprising a form; the weapons 
used seem so childishly simple when compared 
with the far-reaching results; but there is no 
escape from the conclusions. There are no 
complications of detail — one line of attack is 
enough in each case; and this is true too of the 
proofs of many much more difficult theorems, 
the full appreciation of which demands quite 
a high degree of technical proficiency. We 
do not want many 'variations' in the proof 
of a mathematical theorem : ' enumeration of 
cases', indeed, is one of the duller forms of 
mathematical argument. A mathematical 
proof should resemble a simple and clear-cut 
constellation, not a scattered cluster in the 
Milky Way. 


A chess problem also has unexpectedness, 
and a certain economy; it is essential that the 
moves should be surprising, and that every 
piece on the board should play its part. But 
the aesthetic effect is cumulative. It is essential 
also (unless the problem is too simple to be 
really amusing) that the key-move should be 
followed by a good many variations, each 
requiring its own individual answer. ' If P-B5 

then Kt-R6; if then ; if then 

.... ' — the effect would be spoilt if there were 
not a good many different replies. All this is 
quite genuine mathematics, and has its merits ; 
but it is just that 'proof by enumeration of 
cases' (and of cases which do not, at bottom, 
differ at all profoundly*) which a real mathe- 
matician tends to despise. 

I am inclined to think that I could reinforce 
my argument by appealing to the feelings 
of chess-players themselves. Surely a chess 
master, a player of great games and great 
matches, at bottom scorns a problemist's 
purely mathematical art. He has much of it 

* I believe that it is now regarded as a merit in a problem 
that there should be many variations of the same type. 

II 4 

in reserve himself, and can produce it in an 
emergency : ' if he had made such and such a 
move, then I had such and such a winning 
combination in mind.' But the 'great game' 
of chess is primarily psychological, a conflict 
between one trained intelligence and another, 
and not a mere collection of small mathema- 
tical theorems. 


I must return to my Oxford apology, and 
examine a little more carefully some of the 
points which I postponed in § 6. It will be 
obvious by now that I am interested in mathe- 
matics only as a creative art. But there are 
other questions to be considered, and in 
particular that of the 'utility' (or uselessness) 
of mathematics, about which there is much 
confusion of thought. We must also consider 
whether mathematics is really quite so ' harm- 
less' as I took for granted in my Oxford 

A science or an art may be said to be ' useful ' 
if its development increases, even indirectly, 

the material well-being and comfort of men, 
if it promotes happiness, using that word in a 
crude and commonplace way. Thus medicine 
and physiology are useful because they relieve 
suffering, and engineering is useful because it 
helps us to build houses and bridges, and so 
to raise the standard of life (engineering, of 
course, does harm as well, but that is not the 
question at the moment) . Now some mathe- 
matics is certainly useful in this way; the 
engineers could not do their job without a 
fair working knowledge of mathematics, and 
mathematics is beginning to find applications 
even in physiology. So here we have a possible 
ground for a defence of mathematics ; it may 
not be the best, or even a particularly strong 
defence, but it is one which we must examine. 
The 'nobler' uses of mathematics, if such they 
be, the uses which it shares with all creative 
art, will be irrelevant to our examination. 
Mathematics may, like poetry or music, 'pro- 
mote and sustain a lofty habit of mind', and 
so increase the happiness of mathematicians 
and even of other people ; but to defend it on 
that ground would be merely to elaborate 


what I have said already. What we have to 
consider now is the 'crude' utility of mathe- 


All this may seem very obvious, but even 
here there is often a good deal of confusion, 
since the most 'useful' subjects are quite com- 
monly just those which it is most useless for 
most of us to learn. It is useful to have an 
adequate supply of physiologists and en- 
gineers; but physiology and engineering are 
not useful studies for ordinary men (though 
their study may of course be defended on other 
grounds) . For my own part I have never once 
found myself in a position where such scientific 
knowledge as I possess, outside pure mathe- 
matics, has brought me the slightest advan- 

It is indeed rather astonishing how little 
practical value scientific knowledge has for 
ordinary mei, how dull and commonplace 
such of it as has value is, and how its value 
seems almost to vary inversely to its reputed 


utility. It is useful to be tolerably quick at 
common arithmetic (and that, of course, is 
pure mathematics). It is useful to know a 
little French or German, a little history and 
geography, perhaps even a little economics. 
But a little chemistry, physics, or physiology 
has no value at all in ordinary life. We know 
that the gas will burn without knowing its 
constitution; when our cars break down we 
take them to a garage; when our stomach is 
out of order, we go to a doctor or a drugstore. 
We live either by rule of thumb or on other 
people's professional knowledge. 

However, this is a side issue, a matter of 
pedagogy, interesting only to schoolmasters 
who have to advise parents clamouring for a 
1 useful ' education for their sons. Of course 
we do not mean, when we say that physiology 
is useful, that most people ought to study 
physiology, but that the development of 
physiology by a handful of experts will increase 
the comfort of the majority. The questions 
which are important for us now are, how far 
mathematics can claim this sort of utility, 
what kinds of mathematics can make the 


strongest claims, and how far the intensive 
study of mathematics, as it is understood 
by mathematicians, can be justified on this 
ground alone. 


It will probably be plain by now to what 
conclusions I am coming ; so I will state them 
at once dogmatically and then elaborate them 
a little. It is undeniable that a good deal of 
elementary mathematics — and I use the word 
' elementary ' in the sense in which professional 
mathematicians use it, in which it includes, 
for example, a fair working knowledge of the 
differential and integral calculus — has con- 
siderable practical utility. These parts of 
mathematics are, on the whole, rather dull; 
they are just the parts which have least 
aesthetic value. The 'real' mathematics of the 
'real' mathematicians, the mathematics of 
Fermat and Euler and Gauss and Abel and 
Riemann, is almost wholly 'useless' (and this 
is as true of 'applied' as of 'pure' mathe- 
matics). It is not possible to justify the life of 



any genuine professional mathematician on 
the ground of the 'utility' of his work. 

But here I must deal with a misconception. 
It is sometimes suggested that pure mathema- 
ticians glory in the uselessness of their work*, 
and make it a boast that it has no practical 
applications. The imputation is usually based 
on an incautious saying attributed to Gauss, 
to the effect that, if mathematics is the queen 
of the sciences, then the theory of numbers is, 
because of its supreme uselessness, the queen 
of mathematics — I have never been able to find 
an exact quotation. I am sure that Gauss's 
saying (if indeed it be his) has been rather 
crudely misinterpreted. If the theory of num- 
bers could be employed for any practical and 
obviously honourable purpose, if it could be 
turned directly to the furtherance of human 
happiness or the relief of human suffering, as 

* I have been accused of taking this view myself. I once 
said that 'a science is said to be useful if its development 
tends to accentuate the existing inequalities in the distribu- 
tion of wealth, or more directly promotes the destruction of 
human life', and this sentence, written in 1915, has been 
quoted (for or against me) several times. It was of course 
a conscious rhetorical flourish, though one perhaps excusable 
at the time when it was written. 


physiology and even chemistry can, then 
surely neither Gauss nor any other mathema- 
tician would have been so foolish as to decry 
or regret such applications. But science works 
for evil as well as for good (and particularly, 
of course, in time of war) ; and both Gauss and 
lesser mathematicians may be justified in 
rejoicing that there is one science at any rate, 
and that their own, whose very remoteness 
from ordinary human activities should keep 
it gentle and clean. 


There is another misconception against 
which we must guard. It is quite natural to 
suppose that there is a great difference in 
utility between 'pure' and 'applied' mathe- 
matics. This is a delusion: there is a sharp 
distinction between the two kinds of mathe- 
matics, which I will explain in a moment, but 
it hardly affects their utility. 

How do pure and applied mathematics 
differ from one another? This is a question 
which can be answered definitely and about 

which there is general agreement among 
mathematicians. There will be nothing in the 
least unorthodox about my answer, but it 
needs a little preface. 

My next two sections will have a mildly 
philosophical flavour. The philosophy will not 
cut deep, or be in any way vital to my main 
theses; but I shall use words which are used 
very frequently with definite philosophical 
implications, and a reader might well become 
confused if I did not explain how I shall use 

I have often used the adjective 'real', and 
as we use it commonly in conversation. I have 
spoken of ' real mathematics ' and ' real mathe- 
maticians ', as I might have spoken of 'real 
poetry' or 'real poets', and I shall continue 
to do so. But I shall also use the word ' reality ', 
and with two different connotations. 

In the first place, I shall speak of ' physical 
reality', and here again I shall be using the 
word in the ordinary sense. By physical reality 
I mean the material world, the world of day 
and night, earthquakes and eclipses, the world 
which physical science tries to describe. 


I hardly suppose that, up to this point, any 
reader is likely to find trouble with my lan- 
guage, but now I am near to more difficult 
ground. For me, and I suppose for most 
mathematicians, there is another reality, which 
I will call 'mathematical reality'; and there 
is no sort of agreement about the nature of 
mathematical reality among either mathema- 
ticians or philosophers. Some hold that it is 
' mental ' and that in some sense we construct 
it, others that it is outside and independent of 
us. A man who could give a convincing 
account of mathematical reality would have 
solved very many of the most difficult problems 
of metaphysics. If he could include physical 
reality in his account, he would have solved 
them all. 

I should not wish to argue any of these 
questions here even if I were competent to do 
so, but I will state my own position dogma- 
tically in order to avoid minor misapprehen- 
sions. I believe that mathematical reality lies 
outside us, that our function is to discover or 
observe it, and that the theorems which we 
prove, and which we describe grandiloquently 

as our 'creations', are simply our notes of our 
observations. This view has been held, in one 
form or another, by many philosophers of high 
reputation from Plato onwards, and I shall use 
the language which is natural to a man who 
holds it. A reader who does not like the 
philosophy can alter the language : it will make 
very little difference to my conclusions. 


The contrast between pure and applied 
mathematics stands out most clearly, perhaps, 
in geometry. There is the science of pure geo- 
metry*, in which there are many geometries, 
projective geometry, Euclidean geometry, 
non-Euclidean geometry, and so forth. Each 
of these geometries is a model, a pattern of ideas, 
and is to be judged by the interest and beauty 
of its particular pattern. It is a map or picture, 
the joint product of many hands, a partial and 
imperfect copy (yet exact so far as it extends) 

* We must of course, for the purposes of this discussion, 
count as pure geometry what mathematicians call ' analytical ' 


of a section of mathematical reality. But the 
point which is important to us now is this, that 
there is one thing at any rate of which pure 
geometries are not pictures, and that is the 
spatio-temporal reality of the physical world. 
It is obvious, surely, that they cannot be, since 
earthquakes and eclipses are not mathema- 
tical concepts. 

This may sound a little paradoxical to an 
outsider, but it is a truism to a geometer ; and 
I may perhaps be able to make it clearer by 
an illustration. Let us suppose that I am giving 
a lecture on some system of geometry, such as 
ordinary Euclidean geometry, and that I draw 
figures on the blackboard to stimulate the 
imagination of my audience, rough drawings 
of straight lines or circles or ellipses. It is plain, 
first, that the truth of the theorems which I 
prove is in no way affected by the quality of 
my drawings. Their function is merely to bring 
home my meaning to my hearers, and, if I can 
do that, there would be no gain in having them 
redrawn by the most skilful draughtsman. 
They are pedagogical illustrations, not part of 
the real subject-matter of the lecture. 

I2 5 

Now let us go a stage further. The room in 
which I am lecturing is part of the physical 
world, and has itself a certain pattern. The 
study of that pattern, and of the general 
pattern of physical reality, is a science in itself, 
which we may call ' physical geometry'. Sup- 
pose now that a violent dynamo, or a massive 
gravitating body, is introduced into the room. 
Then the physicists tell us that the geometry 
of the room is changed, its whole physical 
pattern slightly but definitely distorted. Do 
the theorems which I have proved become 
false? Surely it would be nonsense to suppose 
that the proofs of them which I have given are 
affected in any way. It would be like sup- 
posing that a play of Shakespeare is changed 
when a reader spills his tea over a page. 
The play is independent of the pages on 
which it is printed, and 'pure geometries' 
are independent of lecture rooms, or of any 
other detail of the physical world. 

This is the point of view of a pure mathe- 
matician. Applied mathematicians, mathe- 
matical physicists, naturally take a different 
view, since they are preoccupied with the 


physical world itself, which also has its structure 
or pattern. We cannot describe this pattern 
exactly, as we can that of a pure geometry, but 
we can say something significant about it. We 
can describe, sometimes fairly accurately, 
sometimes very roughly, the relations which 
hold between some of its constituents, and 
compare them with the exact relations holding 
between constituents of some system of pure 
geometry. We may be able to trace a certain 
resemblance between the two sets of relations, 
and then the pure geometry will become 
interesting to physicists ; it will give us, to that 
extent, a map which 'fits the facts' of the 
physical world. The geometer offers to the 
physicist a whole set of maps from which to 
choose. One map, perhaps, will fit the facts 
better than others, and then the geometry 
which provides that particular map will be 
the geometry most important for applied 
mathematics. I may add that even a pure 
mathematician may find his appreciation of 
this geometry quickened, since there is no 
mathematician so pure that he feels no interest 
at all in the physical world; but, in so far as 


he succumbs to this temptation, he will be 
abandoning his purely mathematical position. 


There is another remark which suggests 
itself here and which physicists may find para- 
doxical, though the paradox will probably 
seem a good deal less than it did eighteen years 
ago. I will express it in much the same words 
which I used in 1922 in an address to Section A 
of the British Association. My audience then 
was composed almost entirely of physicists, 
and I may have spoken a little provocatively 
on that account; but I would still stand by the 
substance of what I said. 

I began by saying that there is probably less 
difference between the positions of a mathe- 
matician and of a physicist than is generally 
supposed, and that the most important seems 
to me to be this, that the mathematician is in 
much more direct contact with reality. This 
may seem a paradox, since it is the physicist 
who deals with the subject-matter usually 
described as 'real' ; but a very little reflection 


is enough to show that the physicist's reality, 
whatever it may be, has few or none of the 
attributes which common sense ascribes in- 
stinctively to reality. A chair may be a 
collection of whirling electrons, or an idea in 
the mind of God : each of these accounts of it 
may have its merits, but neither conforms at 
all closely to the suggestions of common sense. 

I went on to say that neither physicists nor 
philosophers have ever given any convincing 
account of what ' physical reality ' is, or of how 
the physicist passes, from the confused mass of 
fact or sensation with which he starts, to the 
construction of the objects which he calls 
'real'. Thus we cannot be said to know what 
the subject-matter of physics is; but this need 
not prevent us from understanding roughly 
what a physicist is trying to do. It is plain 
that he is trying to correlate the incoherent 
body of crude fact confronting him with some 
definite and orderly scheme of abstract rela- 
tions, the kind of scheme which he can borrow 
only from mathematics. 

A mathematician, on the other hand, is 
working with his own mathematical reality. 

Of this reality, as I explained in § 22, 1 take a 
' realistic ' and not an ' idealistic ' view. At any 
rate (and this was my main point) this realistic 
view is much more plausible of mathematical 
than of physical reality, because mathematical 
objects are so much more what they seem. 
A chair or a star is not in the least like what it 
seems to be ; the more we think of it, the fuzzier 
its outlines become in the haze of sensation 
which surrounds it; but '2' or '317' has 
nothing to do with sensation, and its properties 
stand out the more clearly the more closely we 
scrutinize it. It may be that modern physics 
fits best into some framework of idealistic 
philosophy — I do not believe it, but there are 
eminent physicists who say so. Pure mathe- 
matics, on the other hand, seems to me a rock 
on which all idealism founders: 3 1 7 is a prime, 
not because we think so, or because our minds 
are shaped in one way rather than another, 
but because it is so, because mathematical reality 
is built that way. 



These distinctions between pure and applied 
mathematics are important in themselves, but 
they have very little bearing on our discussion 
of the 'usefulness' of mathematics. I spoke in 
§ 21 of the 'real' mathematics of Fermat and 
other great mathematicians, the mathematics 
which has permanent aesthetic value, as for 
example the best Greek mathematics has, the 
mathematics which is eternal because the best 
of it may, like the best literature, continue to 
cause intense emotional satisfaction to thou- 
sands of people after thousands of years. These 
men were all primarily pure mathematicians 
(though the distinction was naturally a good 
deal less sharp in their days than it is now) ; 
but I was not thinking only of pure mathe- 
matics. I count Maxwell and Einstein, Edding- 
ton and Dirac, among 'real' mathematicians. 
The great modern achievements of applied 
mathematics have been in relativity and 
quantum mechanics, and these subjects are, 
at present at any rate, almost as 'useless' 

I3 1 

as the theory of numbers. It is the dull and 
elementary parts of applied mathematics, as 
it is the dull and elementary parts of pure 
mathematics, that work for good or ill. Time 
may change all this. No one foresaw the 
applications of matrices and groups and other 
purely mathematical theories to modern 
physics, and it may be that some of the 
' highbrow ' applied mathematics will become 
'useful' in as unexpected a way; but the 
evidence so far points to the conclusion that, 
in one subject as in the other, it is what is 
commonplace and dull that counts for prac- 
tical life. 

I can remember Eddington giving a happy 
example of the unattractiveness of 'useful 5 
science. The British Association held a meeting 
in Leeds, and it was thought that the members 
might like to hear something of the applica- 
tions of science to the 'heavy woollen' in- 
dustry. But the lectures and demonstrations 
arranged for this purpose were rather a fiasco. 
It appeared that the members (whether 
citizens of Leeds or not) wanted to be enter- 
tained, and that 'heavy wool' is not at all an 


entertaining subject. So the attendance at 
these lectures was very disappointing; but 
those who lectured on the excavations at 
Knossos, or on relativity, or on the theory 
of prime numbers, were delighted by the 
audiences that they drew. 


What parts of mathematics are useful? 

First, the bulk of school mathematics, 
arithmetic, elementary algebra, elementary 
Euclidean geometry, elementary differential 
and integral calculus. We must except a cer- 
tain amount of what is taught to 'specialists', 
such as projective geometry. In applied mathe- 
matics, the elements of mechanics (electricity, 
as taught in schools, must be classified as 
physics) . 

Next, a fair proportion of university mathe- 
matics is also useful, that part of it which is 
really a development of school mathematics 
with a more finished technique, and a certain 
amount of the more physical subjects such as 
electricity and hydromechanics. We must also 

l 33 

remember that a reserve of knowledge is always 
an advantage, and that the most practical of 
mathematicians may be seriously handicapped 
if his knowledge is the bare minimum which 
is essential to him ; and for this reason we must 
add a little under every heading. But our 
general conclusion must be that such mathe- 
matics is useful as is wanted by a superior 
engineer or a moderate physicist ; and that is 
roughly the same thing as to say, such mathe- 
matics as has no particular aesthetic merit. 
Euclidean geometry, for example, is useful 
in so far as it is dull — we do not want the 
axiomatics of parallels, or the theory of pro- 
portion, or the construction of the regular 

One rather curious conclusion emerges, that 
pure mathematics is on the whole distinctly 
more useful than applied. A pure mathema- 
tician seems to have the advantage on the 
practical as well as on the aesthetic side. For 
what is useful above all is technique, and 
mathematical technique is taught mainly 
through pure mathematics. 

I hope that I need not say that I am not 


trying to decry mathematical physics, a splen- 
did subject with tremendous problems where 
the finest imaginations have run riot. But is 
not the position of an ordinary applied mathe- 
matician in some ways a little pathetic? If he 
wants to be useful, he must work in a humdrum 
way, and he cannot give full play to his fancy 
even when he wishes to rise to the heights. 
'Imaginary' universes are so much more 
beautiful than this stupidly constructed 'real' 
one; and most of the finest products of an 
applied mathematician's fancy must be re- 
jected, as soon as they have been created, for 
the brutal but sufficient reason that they do 
not fit the facts. 

The general conclusion, surely, stands out 
plainly enough. If useful knowledge is, as we 
agreed provisionally to say, knowledge which 
is likely, now or in the comparatively near 
future, to contribute to the material comfort 
of mankind, so that mere intellectual satis- 
faction is irrelevant, then the great bulk of 
higher mathematics is useless. Modern geo- 
metry and algebra, the theory of numbers, the 
theory of aggregates and functions, relativity, 


quantum mechanics — no one of them stands 
the test much better than another, and there 
is no real mathematician whose life can be 
justified on this ground. If this be the test, 
then Abel, Riemann, and Poincare wasted 
their lives; their contribution to human com- 
fort was negligible, and the world would have 
been as happy a place without them. 


I t may be obj ected that my concept of ' utility ' 
has been too narrow, that I have defined it in 
terms of ' happiness ' or 'comfort' only, and 
have ignored the general 'social' effects of 
mathematics on which recent writers, with 
very different sympathies, have laid so much 
stress. Thus Whitehead (who has been a 
mathematician) speaks of 'the tremendous 
effect of mathematical knowledge on the lives 
of men, on their daily avocations, on the 
organization of society'; and Hogben (who 
is as unsympathetic to what I and other 
mathematicians call mathematics as White- 
head is sympathetic) says that 'without a 

knowledge of mathematics, the grammar of 
size and order, we cannot plan the rational 
society in which there will be leisure for all and 
poverty for none ' (and much more to the same 

I cannot really believe that all this eloquence 
will do much to comfort mathematicians. The 
language of both writers is violently exag- 
gerated, and both of them ignore very obvious 
distinctions. This is very natural in Hogben's 
case, since he is admittedly not a mathema- 
tician ; he means by ' mathematics ' the mathe- 
matics which he can understand, and which 
I have called 'school' mathematics. This 
mathematics has many uses, which I have 
admitted, which we can call 'social' if we 
please, and which Hogben enforces with many 
interesting appeals to the history of mathe- 
matical discovery. It is this which gives his 
book its merit, since it enables him to make 
plain, to many readers who never have been 
and never will be mathematicians, that there 
is more in mathematics than they thought. 
But he has hardly any understanding of 'real' 
mathematics (as any one who reads what he 


says about Pythagoras's theorem, or about 
Euclid and Einstein, can tell at once), and 
still less sympathy with it (as he spares no 
pains to show) . ' Real' mathematics is to him 
merely an object of contemptuous pity. 

It is not lack of understanding or of sym- 
pathy which is the trouble in Whitehead's 
case; but he forgets, in his enthusiasm, dis- 
tinctions with which he is quite familiar. The 
mathematics which has this 'tremendous 
effect' on the 'daily avocations of men' and 
on 'the organization of society' is not the 
Whitehead but the Hogben mathematics. The 
mathematics which can be used ' for ordinary 
purposes by ordinary men' is negligible, and 
that which can be used by economists or 
sociologists hardly rises to 'scholarship stan- 
dard'. The Whitehead mathematics may 
affect astronomy or physics profoundly, philo- 
sophy very appreciably — high thinking of one 
kind is always likely to affect high thinking of 
another — but it has extremely little effect on 
anything else. Its 'tremendous effects' have 
been, not on men generally, but on men like 
Whitehead himself. 



There are then two mathematics. There is 
the real mathematics of the real mathema- 
ticians, and there is what I will call the ' trivial ' 
mathematics, for want of a better word. The 
trivial mathematics may be justified by argu- 
ments which would appeal to Hogben, or 
other writers of his school, but there is no such 
defence for the real mathematics, which must 
be justified as art if it can be justified at all. 
There is nothing in the least paradoxical or 
unusual in this view, which is that held com- 
monly by mathematicians. 

We have still one more question to consider. 
We have concluded that the trivial mathe- 
matics is, on the whole, useful, and that the 
real mathematics, on the whole, is not; that 
the trivial mathematics does, and the real 
mathematics does not, ' do good ' in a certain 
sense ; but we have still to ask whether either 
sort of mathematics does harm. It would be 
paradoxical to suggest that mathematics of 
any sort does much harm in time of peace, so 


that we are driven to the consideration of the 
effects of mathematics on war. It is very 
difficult to argue such questions at all dis- 
passionately now, and I should have preferred 
to avoid them; but some sort of discussion 
seems inevitable. Fortunately, it need not be 
a long one. 

There is one comforting conclusion which 
is easy for a real mathematician. Real mathe- 
matics has no effects on war. No one has yet 
discovered any warlike purpose to be served 
by the theory of numbers or relativity, and it 
seems very unlikely that anyone will do so for 
many years. It is true that there are branches 
of applied mathematics, such as ballistics and 
aerodynamics, which have been developed 
deliberately for war and demand a quite 
elaborate technique: it is perhaps hard to call 
them ' trivial', but none of them has any claim 
to rank as 'real'. They are indeed repulsively 
ugly and intolerably dull; even Littlewood 
could not make ballistics respectable, and if he 
could not who can? So a real mathematician 
has his conscience clear; there is nothing to 
be set against any value his work may have ; 

mathematics is, as I said at Oxford, a ■ harm- 
less and innocent' occupation. 

The trivial mathematics, on the other hand, 
has many applications in war. The gunnery 
experts and aeroplane designers, for example, 
could not do their work without it. And the 
general effect of these applications is plain : 
mathematics facilitates (if not so obviously as 
physics or chemistry) modern, scientific, 'total' 

It is not so clear as it might seem that this 
is to be regretted, since there are two sharply 
contrasted views about modern scientific war. 
The first and the most obvious is that the effect 
of science on war is merely to magnify its 
horror, both by increasing the sufferings of the 
minority who have to fight and by extending 
them to other classes. This is the most natural 
and the orthodox view. But there is a very 
different view which seems also quite tenable, 
and which has been stated with great force by 
Haldane in Callinicus*. It can be maintained 
that modern warfare is less horrible than the 

* J. B. S. Haldane, Callinicus: a Defence of Chemical Warfare 


warfare of pre -scientific times; that bombs are 
probably more merciful than bayonets; that 
lachrymatory gas and mustard gas are perhaps 
the most humane weapons yet devised by 
military science; and that the orthodox view 
rests solely on loose-thinking sentimentalism*. 
It may also be urged (though this was not one 
of Haldane's theses) that the equalization of 
risks which science was expected to bring 
would be in the long run salutary; that a 
civilian's life is not worth more than a soldier's, 
nor a woman's than a man's; that anything is 
better than the concentration of savagery on 
one particular class; and that, in short, the 
sooner war comes ' all out ' the better. 

I do not know which of these views is nearer 
to the truth. It is an urgent and a moving 
question, but I need not argue it here. It 
concerns only the ' trivial ' mathematics, which 
it would be Hogben's business to defend rather 
than mine. The case for his mathematics may 

* I do not wish to prejudge the question by this much 
misused word ; it may be used quite legitimately to indicate 
certain types of unbalanced emotion. Many people, of course, 
use ' sentimentalism ' as a term of abuse for other people's 
decent feelings, and 'realism' as a disguise for their own 

I 4 2 

be rather more than a little soiled ; the case for 
mine is unaffected. 

Indeed, there is more to be said, since there 
is one purpose at any rate which the real 
mathematics may serve in war. When the 
world is mad, a mathematician may find in 
mathematics an incomparable anodyne. For 
mathematics is, of all the arts and sciences, the 
most austere and the most remote, and a 
mathematician should be of all men the one 
who can most easily take refuge where, as 
Bertrand Russell says, 'one at least of our 
nobler impulses can best escape from the 
dreary exile of the actual world'. It is a pity 
that it should be necessary to make one very 
serious reservation — he must not be too old. 
Mathematics is not a contemplative but a 
creative subject; no one can draw much con- 
solation from it when he has lost the power or 
the desire to create ; and that is apt to happen 
to a mathematician rather soon. It is a pity, 
but in that case he does not matter a great deal 
anyhow, and it would be silly to bother about 


2 9 

I wi l l end with a summary of my conclusions, 
but putting them in a more personal way. 
I said at the beginning that anyone who de- 
fends his subject will find that he is defending 
himself; and my justification of the life of a 
professional mathematician is bound to be, at 
bottom, a justification of my own. Thus this 
concluding section will be in its substance a 
fragment of autobiography. 

I cannot remember ever having wanted to 
be anything but a mathematician. I suppose 
that it was always clear that my specific 
abilities lay that way, and it never occurred to 
me to question the verdict of my elders. I do 
not remember having felt, as a boy, any 
passion for mathematics, and such notions as I 
may have had of the career of a mathematician 
were far from noble. I thought of mathe- 
matics in terms of examinations and scholar- 
ships: I wanted to beat other boys, and this 
seemed to be the way in which I could do so 
most decisively. 


I was about fifteen when (in a rather odd 
way) my ambitions took a sharper turn. There 
is a book by 'Alan St Aubyn'* called A Fellow 
of Trinity, one of a series dealing with what is 
supposed to be Cambridge college life. I sup- 
pose that it is a worse book than most of Marie 
Corelli's; but a book can hardly be entirely 
bad if it fires a clever boy's imagination. There 
are two heroes, a primary hero called Flowers, 
who is almost wholly good, and a secondary 
hero, a much weaker vessel, called Brown. 
Flowers and Brown find many dangers in 
university life, but the worst is a gambling 
saloon in Chesterton | run by the Misses 
Bellenden, two fascinating but extremely 
wicked young ladies. Flowers survives all 
these troubles, is Second Wrangler and Senior 
Classic, and succeeds automatically to a Fel- 
lowship (as I suppose he would have done 
then). Brown succumbs, ruins his parents, 
takes to drink, is saved from delirium tremens 
during a thunderstorm only by the prayers of 

* 'Alan St Aubyn' was Mrs Frances Marshall, wife of 
Matthew Marshall. 

t Actually, Chesterton lacks picturesque features. 


the Junior Dean, has much difficulty in ob- 
taining even an Ordinary Degree, and ulti- 
mately becomes a missionary. The friendship 
is not shattered by these unhappy events, and 
Flowers's thoughts stray to Brown, with 
affectionate pity, as he drinks port and eats 
walnuts for the first time in Senior Combina- 
tion Room. 

Now Flowers was a decent enough fellow 
(so far as 'Alan St Aubyn' could draw one), 
but even my unsophisticated mind refused to 
accept him as clever. If he could do these 
things, why not I? In particular, the final 
scene in Combination Room fascinated me 
completely, and from that time, until I ob- 
tained one, mathematics meant to me primarily 
a Fellowship of Trinity. 

I found at once, when I came to Cambridge, 
that a Fellowship implied 'original work', but 
it was a long time before I formed any definite 
idea of research. I had of course found at 
school, as every future mathematician does, 
that I could often do things much better than 
my teachers; and even at Cambridge I found, 
though naturally much less frequently, that 


I could sometimes do things better than the 
College lecturers. But I was really quite 
ignorant, even when I took the Tripos, of 
the subjects on which I have spent the rest of 
my life ; and I still thought of mathematics 
as essentially a ' competitive ' subject. My eyes 
were first opened by Professor Love, who taught 
me for a few terms and gave me my first 
serious conception of analysis. But the great 
debt which I owe to him — he was, after all, 
primarily an applied mathematician — was his 
advice to read Jordan's famous Cours a" analyse; 
and I shall never forget the astonishment with 
which I read that remarkable work, the first 
inspiration for so many mathematicians of my 
generation, and learnt for the first time as 
I read it what mathematics really meant. 
From that time onwards I was in my way 
a real mathematician, with sound mathe- 
matical ambitions and a genuine passion for 

I wrote a great deal during the next ten 
years, but very little of any importance; there 
are not more than four or five papers which I 
can still remember with some satisfaction. The 


real crises of my career came ten or twelve 
years later, in 191 1, when I began my long 
collaboration with Little wood, and in 1913, 
when I discovered Ramanujan. All my best 
work since then has been bound up with theirs, 
and it is obvious that my association with them 
was the decisive event of my life. I still say to 
myself when I am depressed, and find myself 
forced to listen to pompous and tiresome 
people, ' Well, I have done one thing you could 
never have done, and that is to have col- 
laborated with both Littlewood and Rama- 
nujan on something like equal terms.' It is to 
them that I owe an unusually late maturity : I 
was at my best at a little past forty, when I was a 
professor at Oxford. Since then I have suffered 
from that steady deterioration which is the 
common fate of elderly men and particularly 
of elderly mathematicians. A mathematician 
may still be competent enough at sixty, but it 
is useless to expect him to have original ideas. 
It is plain now that my life, for what it is 
worth, is finished, and that nothing I can do 
can perceptibly increase or diminish its value. 
It is very difficult to be dispassionate, but I 


count it a ' success ' ; I have had more reward 
and not less than was due to a man of my 
particular grade of ability. I have held a 
series of comfortable and ' dignified ' positions. 
I have had very little trouble with the duller 
routine of universities. I hate 'teaching', and 
have had to do very little, such teaching 
as I have done having been almost entirely 
supervision of research ; I love lecturing, and 
have lectured a great deal to extremely able 
classes; and I have always had plenty of 
leisure for the researches which have been the 
one great permanent happiness of my life. 
I have found it easy to work with others, and 
have collaborated on a large scale with two 
exceptional mathematicians; and this has 
enabled me to add to mathematics a good deal 
more than I could reasonably have expected. 
I have had my disappointments, like any other 
mathematician, but none of them has been too 
serious or has made me particularly unhappy. 
If I had been offered a life neither better nor 
worse when I was twenty, I would have 
accepted without hesitation. 

It seems absurd to suppose that I could have 

J 49 

'done better'. I have no linguistic or artistic 
ability, and very little interest in experimental 
science. I might have been a tolerable philo- 
sopher, but not one of a very original kind. 
I think that I might have made a good lawyer ; 
but journalism is the only profession, outside 
academic life, in which I should have felt really 
confident of my chances. There is no doubt that 
I was right to be a mathematician, if the cri- 
terion is to be what is commonly called success. 

My choice was right, then, if what I wanted 
was a reasonably comfortable and happy life. 
But solicitors and stockbrokers and book- 
makers often lead comfortable and happy 
lives, and it is very difficult to see how the 
world is the richer for their existence. Is there 
any sense in which I can claim that my life has 
been less futile than theirs? It seems to me 
again that there is only one possible answer : 
yes, perhaps, but, if so, for one reason only. 

I have never done anything 'useful'. No 
discovery of mine has made, or is likely to 
make, directiy or indirectly, for good or ill, 
the least difference to the amenity of the 
world. I have helped to train other mathema- 


ticians, but mathematicians of the same kind 
as myself, and their work has been, so far at 
any rate as I have helped them to it, as useless 
as my own. Judged by all practical standards, 
the value of my mathematical life is nil ; and 
outside mathematics it is trivial anyhow. I 
have just one chance of escaping a verdict of 
complete triviality, that I may be judged to 
have created something worth creating. And 
that I have created something is undeniable : 
the question is about its value. 

The case for my life, then, or for that of any 
one else who has been a mathematician in the 
same sense in which I have been one, is this : 
that I have added something to knowledge, 
and helped others to add more ; and that these 
somethings have a value which differs in 
degree only, and not in kind, from that of the 
creations of the great mathematicians, or of 
any of the other artists, great or small, who 
have left some kind of memorial behind 

I5 1 


Professor Broad and Dr Snow have both remarked 
to me that, if I am to strike a fair balance between 
the good and evil done by science, I must not allow 
myself to be too much obsessed by its effects on 
war; and that, even when I am thinking of them, 
I must remember that it has many very important 
effects besides those which are purely destructive. 
Thus (to take the latter point first), I must re- 
member (a) that the organization of an entire 
population for war is only possible through scientific 
methods; (b) that science has greatly increased 
the power of propaganda, which is used almost 
exclusively for evil; and (c) that it has made 
'neutrality' almost impossible or unmeaning, so 
that there are no longer 'islands of peace' from 
which sanity and restoration might spread out 
gradually after war. All this, of course, tends to 
reinforce the case against science. On the other 
hand, even if we press this case to the utmost, it is 
hardly possible to maintain seriously that the evil 
done by science is not altogether outweighed by 
the good. For example, if ten million lives were 
lost in every war, the net effect of science would 
still have been to increase the average length of 
life. In short, my § 28 is much too 'sentimental'. 


I do not dispute thejustice of these criticisms, but, 
for the reasons which I state in my preface, I have 
found it impossible to meet them in my text, and 
content myself with this acknowledgement. 

Dr Snow has also made an interesting minor 
point about § 8. Even if we grant that 'Archimedes 
will be remembered when Aeschylus is forgotten', 
is not mathematical fame a little too ' anonymous ' 
to be wholly satisfying? We could form a fairly 
coherent picture of the personality of Aeschylus 
(still more, of course, of Shakespeare or Tolstoi) 
from their works alone, while Archimedes and 
Eudoxus would remain mere names. 

Mr J. M. Lomas put this point more picturesquely 
when we were passing the Nelson column in 
Trafalgar Square. If I had a statue on a column in 
London, would I prefer the column to be so high 
that the statue was invisible, or low enough for the 
features to be recognizable? I would choose the 
first alternative, Dr Snow, presumably, the second. 


GREATmamemaucians rarely 
write about themselves or 
about their work, and few of 
them would have the literary 
gift to compose an essay of such 
charm, candour, and insight as 
G. H. Hardy's A Mathema- 
tician's Apology. This exquisite 
little book first appeared in 
1940, when Hardy was 63 years 
of age, but it has been out of 
print since 1951 and is now 
being reissued with an exten- 
sive biographical foreword by 
C. P. Snow. 

Twenty years after his death 
Hardy has become an historical 
figure. He was the founder and 
a brilliant member of the 
British school of classical analy- 
sis which flourished in our 
universities up to the time of 
the Second World War. Hardy 
was the perfect example of a 
pure mathematician. To him, 
" real " — that is, pure — mathe- 
matics is the art of making 
patterns of thought, whose value 
is judged by their beauty, 
depth, and seriousness. Those 
branches of mathematics which 
by ordinary standards are con- 
sidered useful for good or evil 
purposes are for the most part 
dull and trivial. This descrip- 
tion of a mathematician differs 
markedly from that assumed by 
the general public, yet its 
truth will be readily affirmed 
by the majority of creative 

Mathematics Is a young man's 
pursuit ; and when Hardy's 
power for original research 
began to decline, sadness and 
resignation came to his life, 
much of which is reflected in 
the pages of this book. He sets 
little store by teaching and 
exposition, which he asserts is 
the business of second-rate 
minds. Yet he tells us that it 
was the study of the cours 
d'analyse of the great French 
mathematician C. Jorden that 
opened his eyes to " real " 
mathematics, and he makes no 
mention of his own " Pure 
Mathematics " which for the 
past 60 years has made a more 
powerful impact on the minds 
of young students and bright 
schoolboys than any mathe- 
matical text in the language. 

C. P. Snow, who knew Hardy 
intimately during the last 
period of his life, traces the 
human side of the story with 
sympathy and affection. He 
relates Hardy's passion for 
cricket, his endearing eccentri- 
cities, and his brilliant conver- 
sation, set against the back- 
ground of college life. 

In its new form the book 
gives a vivid portrait of an 
eminent mathematician and a 
manifesto for mathematics 
itself ijwcuv LwU/i^a*^ « 

"Mathematics, Professor Hardy writes, is a 
young man's game : a creative and not a con- 
templative art, and so, having passed the age 
of 60, he writes this apology for a life spent on 
what the layman wrongly considers the coldest 
of intellectual heights. How wrongly he will 
learn from Professor Hardy in this moving, 
exciting, beautifully written essay. There is 
nothing here which the layman cannot under- 
stand except possibly one theorem, and I know 
no writing — except perhaps Henry James's in- 
troductory essays— which conveys so clearly 
and with such an absence of fuss the excite- 
ment of the creative artist." 

— Graham Greene in the Spectator 

"His little book is exquisitely written, and will 
become a classical statement of the mathema- 
tician's faith." — The Listener 

"Hardy's sardonic confession of how he ever 
came. to be a professional mathematician 
may be specially commended to solemn 
young men who believe they have a call to 
preach the higher arithmetic to mathemati- 
cal infidels " —Scientific Monthly 


Bentley House, 200 Euston Road, London N.W.I 
American Branch: 32 East 57th Street, New York, N.Y. 10022 


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